Gordon’s Conjectures 1 and 2: Pontryagin–van Kampen duality in the hyperfinite setting

Using the ideas of E. I. Gordon we present and further advance an approach, based on nonstandard analysis, to simultaneous approximations of locally compact abelian groups and their duals by (hyper)finite abelian groups, as well of the Haar measures on them. Combining the techniques of nonstandard analysis and the Fourier analytic methods of additive combinatorics we prove the first two of the three Gordon’s Conjectures which were open since 1991 and are crucial both in the formulations and proofs of the approximation theorems for LCA groups and for the Fourier transform. 2010 Mathematics Subject Classification 22B05 (primary); 03H05, 26E35, 28E05, 54J05, 65T50 (secondary)


Introduction
Locally compact abelian groups (briefly, LCA groups) and the Pontryagin-van Kampen Duality Theorem provide a general background on which all the particular instances of the (commutative) Fourier transform can be treated in a uniform way (see, eg, Hewitt and Ross [24,25], Loomis [30], and Rudin [42]). At the same time, the existence of very fast algorithms for the Fourier transform of functions (n-dimensional vectors) f : Z n → C and, more generally, of functions f : G → C defined on an arbitrary finite abelian group G, calls for their systematic employment in approximating the Fourier transforms on various functional spaces over general LCA groups. This is indeed the case in many important particular cases, including the Fourier series of periodic functions f : R → C with a fixed period T > 0, the Fourier transform of functions f : R → C and f : R n → C, the semidiscrete Fourier transform of sequences f : Z → C, etc (see, eg, Epstein [11], Gröchenig [21], and Terras [45]).
The general situation requires us to elaborate the concepts and techniques of approximation of arbitrary LCA groups by finite abelian groups in the first step. In the second IMG group triplet (G, G 0 , G f ) with a hyperfinite abelian ambient group G. In standard terms this means that every LCA group G can be "arbitrarily well" approximated by finite abelian groups in a sense made precise by a definition introduced by Gordon [17] (see Section 1.3, particularly Proposition 1.3.7, Theorem 1.3.10, and -mainly -Corollary 1.3.11 for more details). For compact metrizable groups this approximability concept coincides with that already studied by Alan Turing [46] in the late 1930s (see the second remark following the proof of Proposition 1.3.7).
For an LCA group G, its dual group G = Hom c (G, T) consists of all continuous homomorphisms (characters) γ γ γ : G → T, where T denotes the compact multiplicative group of complex units. Then G, endowed with the componentwise multiplication and the compact-open topology becomes an LCA group again. The Pontryagin-van Kampen Duality Theorem states that the canonical mapping G → G, sending any x x x ∈ G to the character x x x : G → C of the dual group G, given by the pairing x x x(γ γ γ) = γ γ γ(x x x) on G × G, is an isomorphism of topological groups (see, eg, Dikranjan and Stoyanov [9], Hewitt and Ross [24], Morris [36], Pontryagin [38] or Rudin [42]).
For a finite abelian group G, its dual group G = Hom(G, T) consists of all homomorphisms γ : G → T, and it is isomorphic (though not canonically) to G. The fact that the canonical mapping G → G is an isomorphism of (discrete) abelian groups follows immediately. For a hyperfinite abelian group G we denote by G = Hom(G, * T) the set of all internal homomorphisms (characters) γ : G → * T. According to the Transfer Principle, G with the pointwise multiplication is a hyperfinite abelian group internally (though not canonically) isomorphic to G, and the canonical mapping G → G is an internal group isomorphism.
Assuming that (G, G 0 , G f ) is a condensing IMG group triplet with a hyperfinite abelian ambient group G, let us denote by G = G = G f /G 0 its observable trace. Then its dual G is an LCA group, too, hence it is isomorphic to the observable trace H = H f /H 0 of some condensing IMG group triplet (H, H 0 , H f ) with a hyperfinite abelian ambient group H . One can naturally expect that the triplet representing the dual group G can be constructed from the original triplet (G, G 0 , G f ) in some canonical way. The starting point is the natural requirement that H = G. Then the subsets of the dual group G, called the infinitesimal annihilators of the subgroups G 0 , G f , respectively, are subgroups of G. It can be shown that G, G f , G 4

P Zlatoš
The characters γ ∈ G f are infinitesimally close to 1 on the whole subgroup G f of finite elements of G. They play the role of infinitesimal characters. As every γ ∈ G is a group homomorphism, it belongs to G 0 if and only if it is S-continuous as a mapping γ : G → * T. The elements of G 0 play the role of finite characters.
The condensing IMG group triplet G, G f , G 0 will be called the dual triplet of the IMG group triplet (G, G 0 , G f ).
For an internal character γ ∈ G 0 , its observable trace γ can be considered as the element G f γ of the quotient G 0 /G f = G , as well as the observable trace γ of the S-continuous mapping γ : G → * T, ie for x ∈ G f . That way γ : G → T is a continuous character of the LCA group G (see Section 1.3). The assignment γ → γ , depicted in the commutative diagram is a group homomorphism G 0 → G . Its kernel is the subgroup G f ⊆ G 0 of all infinitesimal characters in G. Thus the assignment γ → γ induces an injective group homomorphism G → G from the observable trace G = G 0 /G f of the dual triplet G, G f , G 0 into the dual group G = G f /G 0 of the observable trace G = G f /G 0 of the original triplet (G, G 0 , G f ). The canonical injective homomorphism G 0 /G f → G f /G 0 justifies the identification of the "two observable traces" G f γ and γ . As proved by Gordon in [17] (see also [18]), the assignment G f γ → γ is an isomorphism of the topological group G = G 0 /G f onto a closed subgroup of the topological group G = G f /G 0 .
It is both tempting and natural to conjecture that the canonical mapping G → G is also surjective, ie, that it is an isomorphism of topological groups. This is indeed the first of Gordon's Conjectures from [17] (see also [18, page 132]), which we will prove as our Theorem 2.2.4.
Gordon's Conjecture 1 Let (G, G 0 , G f ) be a condensing IMG group triplet with a hyperfinite abelian ambient group G. Then the canonical mapping G 0 /G The second dual G of the hyperfinite abelian group G can be naturally identified with the original group G. Then the second dual of the original triplet (G, G 0 , G f ) is defined as the condensing IMG triplet G, G 0 , G f . As we shall see in Sections 2.2 and 2.3, Gordon's Conjecture 1 is equivalent to the following statement, which we will prove as our Theorem 2.2.5.
The Triplet Duality Theorem Let (G, G 0 , G f ) be a condensing IMG group triplet with a hyperfinite abelian ambient group G. Then: Given a condensing IMG group triplet (G, G 0 , G f ) with a hyperfinite ambient group G, a positive number d ∈ * R is called a normalizing multiplier or a normalizing coefficient for (G, G 0 , G f ) if the number d |A| is finite and noninfinitesimal for some (or, equivalently, for each) internal set A between G 0 and G f . In particular, for any internal set A between G 0 and G f , d = 1/ |A| is a normalizing coefficient for (G, G 0 , G f ). 6 P Zlatoš for f , g ∈ * C G . In particular, the Fourier transform of a function f ∈ * C G has to be defined by for γ ∈ G. Then, in order to get the Fourier inversion formula and the Plancherel identity, we need to normalize the inner product and the Fourier transform on the space * C G of all internal functions G → * C by means of the coefficient d = 1/d |G|.
In view of the canonical isomorphism of the observable trace G = G 0 /G f and the dual group G = G f /G 0 , there naturally arises the question whether such a d is indeed a normalizing multiplier for the dual triplet G, G f , G 0 . In that case (and only in that case) the measure m m m d , obtained by pushing down the Loeb measure λ d from G to the observable trace G ∼ = G , will be a Haar measure on G . The second of Gordon's conjectures, proved as our Theorem 2.3.4, states that the response to this question is affirmative.
Gordon's Conjecture 2 Let (G, G 0 , G f ) be a condensing IMG group triplet with a hyperfinite abelian ambient group G. Then, for any internal set D such that G 0 ⊆ D ⊆ G f , d = 1/|D| is a normalizing multiplier for (G, G 0 , G f ) and d = |D|/|G| is a normalizing multiplier for the dual triplet G, G f , G 0 . More generally, if d is any normalizing multiplier for (G, G 0 , G f ) then d = (d |G|) −1 is a normalizing multiplier The methods of nonstandard analysis were applied for the first time in the study of the Fourier series of functions T → C by Luxemburg [32]. His key idea consisted in embedding the group of integers Z into the hyperfinite cyclic group Z n , where n ∈ * N N, and an infinitesimal approximation of the group of complex units T by the hyperfinite subgroup {e 2πik/n : k ∈ Z n } ∼ = Z n of its nonstandard extension * T. The first treatment of abstract (commutative) harmonic analysis by nonstandard methods in full generality is due to Gordon. In a series of works culminating in [17,18] he elaborated a nonstandard approach to approximations of LCA groups by hyperfinite abelian groups, formulated a version of Pontryagin-van Kampen duality for them and developed an approach to the approximation of the classical Fourier-Plancherel transform L 2 (G) → L 2 G by the discrete Fourier transform on the approximating hyperfinite group. At the same time, he formulated three fairly fundamental conjectures in [17] which remained open until 2012. In fact, Gordon accepted one not very essential restriction: having worked in a nonstandard universe which was only assumed to be ℵ 1 -saturated, he was forced to deal just with triplets (G, G 0 , G f ) in which G 0 and G f P Zlatoš

Nonstandard analysis
The reader is assumed to have some basic acquaintance with nonstandard analysis, including the nonstandard approach to topology and continuity in terms of monads and equivalence relations of infinitesimal nearness, and the Loeb measure construction.

General setting
Our exposition takes place in a nonstandard universe * V which is an elementary extension of a superstructure V over some set of individuals containing at least all (classical) complex numbers and the elements of the topological space or topological group, as well as index sets, etc, dealt with.
In particular, every standard mathematical (first-order) structure A ∈ V is embedded into its nonstandard extension * A ∈ * V via the mapping a → * a : A → * A such that, for any formula Φ(x 1 , . . . , x n ) in the language of A and elements a 1 , . . . , a n ∈ A, Φ(a 1 , . . . , a n ) is satisfied in A if and only if Φ( * a 1 , . . . , * a n ) is satisfied in * A (the Transfer Principle). Whenever there threatens no confusion we tend to identify a ∈ A with * a ∈ * A, and to denote the corresponding operations and relations in A and * A by the same sign, dropping * in the latter. Similarly, to a function f : A → B in V there (functorially) corresponds a function * f : * A → * B in * V, etc. The nonstandard universe * V can be constructed as a certain ultrapower of the superstructure V with respect to some properly chosen ultrafilter.
In particular, we have the structures of hypernatural numbers * N, hyperintegers * Z, hyperrational numbers * Q, hyperreal numbers * R, and hypercomplex numbers * C with the usual (and, possibly, some additional) operations and relations, extending the structures of natural numbers N, integers Z, rational numbers Q, real numbers R, and complex numbers C, respectively. 9 some uncountable cardinal κ or even polysaturated, ie κ-saturated for some κ bigger than the cardinality of any set in the original (standard) universe V. However, for the sake of generality, we do not specify the saturation degree κ explicitly. Instead we use the term a set or a system of admissible size referring to (external) subsets of the nonstandard universe with the (external) cardinality < κ, and assume that the universe * V is sufficiently saturated, meaning that S = ∅ for any system of internal sets S ⊆ * V of admissible size with the finite intersection property. "Sufficiently saturated" universes can be constructed as ultrapowers with respect to "sufficiently good" ultrafilters (see Chang and Keisler [6]). For most applications, however, an ℵ 1 -saturated nonstandard universe (ie, κ = ℵ 1 ) is sufficient; in that case a system of admissible size is simply a countable one.
Internal sets A which can be put into a one-to-one correspondence via an internal bijection with sets of the form {1, . . . , n} for some n ∈ * N are called hyperfinite; in that case n = |A| is referred to as the number of elements of A. Hyperfinite sets (briefly, HF sets) behave within the internal context much like finite sets, though for n ∈ * N N they are (externally) infinite.
Internal sets, particularly hyperfinite sets, are the simplest objects in the descriptive settheoretic hierarchy. Next to them in this hierarchy there are the galactic or Σ 0 1 (κ)-sets and the monadic or Π 0 1 (κ)-sets, ie, unions and intersections, respectively, of collections of internal sets of admissible size. From the formal point of view, it is these sets which enable the techniques of Nonstandard Analysis based on saturation. In particular, if S is any collection of admissible size with the finite intersection property consisting of monadic sets X ⊆ * V then S = ∅.
An example of a galactic set is the set N of all (standard) natural numbers, in other words, the set of all finite elements of the internal set of all hypernatural numbers * N. Then the infinite hypernatural numbers form the monadic set * N ∞ = * N N.
Further typical and fairly important examples of monadic sets are the equivalence relations of indistinguishability or infinitesimal nearness, arising in nonstandard models of metric, uniform and (sufficiently regular) topological spaces, as well as the monads (ie, cosets) of particular elements with respect to such equivalence relations.
Every function f : X → Y is considered to be equal to the set of ordered pairs The following lemma describes an important class of relations admitting internal choice functions. P Zlatoš Lemma 1.1.1 (The Internal Choice Lemma) Let X , Y be internal sets in a sufficiently saturated nonstandard universe, and R ⊆ X × Y be a relation such that for every internal set D ⊆ dom R the restriction R D is a monadic set. Then for every galactic set A ⊆ dom R there exists an internal choice function f from R on A.
Sketch of proof If A is internal then the monadic relation R A can be written as the intersection R A = i∈I R i of admissibly many internal relations R i ⊆ X × Y with common domain A. Then, for any nonempty finite set J ⊆ I , we readily obtain an internal choice function f J on A from the relation i∈J R i , by applying the Transfer Principle to the Axiom of Choice. (For hyperfinite A the Axiom of Choice is not even needed at this point.) The existence of an internal choice function f from R on A follows as a consequence of saturation.
If A is a galactic set then the desired conclusion follows from the internal case by applying the saturation argument once again.

Bounded monadic spaces
For the notion of a uniform space and related notions like uniformity, entourage, uniform continuity, etc, we refer the reader to the monograph by Engelking [10].
If (X, U) is a uniform space then the relation of infinitesimal nearness or indistinguishability defined by is a monadic equivalence relation on the nonstandard extension * X of X (provided the respective nonstandard universe is sufficiently saturated). Then the uniformity U consists precisely of those subsets U of X × X satisfying E U ⊆ * U. Instead of (x x x, y y y) ∈ E U we usually write x x x ≈ U y y y or just x x x ≈ y y y for x x x, y y y ∈ * X.
An element y y y ∈ * X is called nearstandard if y y y ≈ U x x x for some x x x ∈ X; the set of all nearstandard elements of * X is denoted by Ns( * X). If the original uniform space (X, U) is Hausdorff (ie, if U is the diagonal Id X ) then there is a well defined standard part mapping Ns( * X) → X sending every y y y ∈ Ns( * X) to its standard part st y y y = • y y y which is the unique element x x x ∈ X such that y y y ≈ x x x. Then the original uniform space (X, U) can be reconstructed as the quotient X ∼ = Ns( * X)/≈ (see the proof of Proposition 1.2.2).
In view of the works by Luxemburg [31], Henson [22] and Goldbring [15,16] we accept as a fact that there is no canonical way how to define the finite elements in the nonstandard extension of a uniform space. Instead of looking for their adequate definition in terms of the uniform structure and standard elements we will treat the equivalence relation of infinitesimal nearness or indistinguishability arising from the uniformity on one hand, and the property of finiteness or accessibility on the other hand as almost independent or just loosely related phenomena to which there correspond different primitive concepts. Our basic nonstandard objects, by means of which we will study (sufficiently regular) topological spaces, will be ordered triples of the form (X, E, X f ) where X is an internal set, E is a monadic equivalence relation on X and X f is a galactic subset of X which is E-closed, ie, x ∈ X f and (x, y) ∈ E imply y ∈ X f , for any x, y ∈ X . We will call them alternatively bounded monadic spaces or IMG spaces like in Ziman and Zlatoš [47], or IMG triplets, indicating that we do not consider the present terminology as definitive.
Intuitively, X is viewed as the underlying or ambient set of the triplet, E is the relation of indistinguishability or infinitesimal nearness on X , and X f is the set of elements of X encompassed by some observational horizon. The elements of X f will be briefly referred to as the finite or accessible ones. To stress the role of the equivalence E we will preferably write x ≈ E y or just x ≈ y instead of (x, y) ∈ E, for x, y ∈ X , and call the set E[x] = {y ∈ X : y ≈ x} of points indistinguishable from the point x ∈ X the E-monad or just the monad of x. The E-closedness of X f can be now expressed as the condition E[x] ⊆ X f for any x ∈ X f . The quotient x ∈ X f } is called the observable trace of the triplet (X, E, X f ).
The restricted quotient mapping X f → X f /E reminds of the standard part mapping Ns( * X) → X in nonstandard extensions of Hausdorff uniform spaces. In order to underline this analogy (especially when viewing the monads as individual points and forgetting about their "sethood") we introduce the notation E[x] = x for the monad of x ∈ X , and A = {a : a ∈ A ∩ X f } for the observable trace of any set A ⊆ X . In particular, the observable trace X = X f = X f /E of the triplet (X, E, X f ) should not be confused with the full quotient X/E ⊇ X . Conversely, for any Y ⊆ X , we call the following set the pretrace of Y: Given an IMG triplet (X, E, X f ), it is an easy exercise in saturation to show that for each internal relation R ⊇ E on X there is a symmetric internal relation S ⊇ E on X such P Zlatoš that S • S ⊆ R. Similarly, for any internal set A ⊆ X f there is an internal set B ⊆ X f and a symmetric internal relation S ⊇ E on X such that S[A] ⊆ B. It follows that there is a downward directed system R of reflexive and symmetric internal relations on X , and an upward directed system B of internal subsets of X , both of admissible size, satisfying the following conditions: Then R becomes a base of a uniformity U E on X (non-Hausdorff, unless E = Id X ). Another base for this uniformity (though not necessarily of admissible size) is formed by all the internal relations R on X such that The observable traces of internal relations R ∈ U E (or just R ∈ R) form a uniformity base on the observable trace X = X f /E, inducing a Hausdorff completely regular topology on it.
We are particularly interested in representing Hausdorff locally compact spaces as observable traces of IMG triplets (X, E, X f ) with a hyperfinite ambient set X . To this end we introduce some types of indices of internal sets A ⊆ X with respect to reflexive and symmetric internal relations S ⊆ X × X : (a) The covering index or entropy of A with respect to S, denoted by A : S , is the least n ∈ * N, such that A ⊆ S[F] for some hyperfinite sets F ⊆ X with n elements, or the symbol ∞ if there is no such n.
(b) The inner covering index of A with respect to S, denoted by A : S i , is the least n ∈ * N, such that A ⊆ S[F] for some hyperfinite sets F ⊆ A with n elements, or the symbol ∞ if there is no such n.
(c) The independence index or the capacity of A with respect to S denoted by A : S , is the biggest n ∈ * N such that there is an n-element set F ⊆ A satisfying (x, y) / ∈ S for any distinct x, y ∈ F , or the symbol ∞ if there is no biggest n with that property.
Then we have the following obvious inequalities (cf Roe [40]): If G is an internal group then, instead of the symmetric and reflexive internal relation S on G, we can take a symmetric internal subset S ⊆ G containing the unit element 1 ∈ G. The obvious modification of the above definitions and the last inequalities to this situation is left to the reader.  x, y ∈ H such that x ≈ y.
Proof (i) ⇔ (ii) is plain, especially for the index A : S i .
(ii) ⇒ (iii) Let R and B be systems of admissible size consisting of internal symmetric relations R ⊆ X × X and internal sets B ⊆ X , respectively, such that E = R and X f = B . Condition (ii) with the index A : S i implies that there is a system of finite sets {F RB : R ∈ R, B ∈ B} such that for all R ∈ R, B ∈ B . Then the set Q = R,B F RB ⊆ X f is of admissible size and, obviously, X f ⊆ S[Q] for any internal relation S ⊇ E on X . Now, it suffices to take for P any subset of Q containing exactly one point from every E-monad intersecting Q.
(iii) ⇒ (iv) Let P be the set guaranteed by (ii) and n be an arbitrary infinite hypernatural number. Then, for each finite set F ⊆ P, we have |F| ≤ n. By saturation, there is a hyperfinite set H such that P ⊆ H ⊆ X and |H| ≤ n. It can be easily verified that (iv) ⇒ (v) Let H ⊆ X f be any infinite hyperfinite set and n be an infinite hypernatural number such that n < |H|. By (iv) there is a hyperfinite set J ⊆ X such that |J| ≤ n and P Zlatoš As |J| < |H|, there is at least one element z ∈ J such that the set H ∩ E[z] contains at least two distinct elements x, y. Then x ≈ y.
(v) ⇒ (ii) Assume that the independence index A : S is not finite for some internal set A ⊆ X f and an internal symmetric relation S ⊇ E. Then, by saturation, there is an infinite hyperfinite set H ⊆ A such that x ≈ y for any distinct x, y ∈ H , contradicting (v).
Remark Let us turn the reader's attention to the following two details. First, in order to prove the implication (iii) ⇒ (iv) the set P just has to be of admissible size and satisfy the condition X f ⊆ S[P] for every internal relation S ⊇ E; it is not necessary that it consists of pairwise distinguishable points. Second, (v) could be strengthened to the following Ramsey type statement: every infinite hyperfinite set H ⊆ X f contains an infinite hyperfinite subset H 0 such that x ≈ y for any x, y ∈ H 0 . Then the implication (iv) ⇒ (v) could still be proved by picking an n ∈ * N ∞ such that, say, n 2 ≤ |H|.
The last condition (v) suggests calling the IMG triplets satisfying any (hence all) of the above conditions condensing (cf Mlček and Zlatoš [34,35] as well as Ziman and Zlatoš [47]). Obviously, the observable trace of any condensing IMG space is locally compact, and the compact subsets of X are exactly the observable traces A of internal subsets A ⊆ X f . However, it should be kept in mind that this is a considerably stronger condition than just the local compactness of X . Nevertheless, we still have the following representation theorem.
Proposition 1.2.2 Let X be a Hausdorff locally compact topological space. Then, in every sufficiently saturated nonstandard universe, there is a condensing IMG triplet (X, E, X f ) such that X is homeomorphic to the observable trace X . If desirable, one can additionally arrange that the ambient space X be hyperfinite.
Proof Let U be some uniformity inducing the topology of X and κ be an uncountable cardinal bigger than the minimal cardinality of some base of U as well as of some open cover of X by relatively compact sets. Let us embed X into its nonstandard extension * X in some κ-saturated nonstandard universe and denote E U = U∈U * U. It can be easily verified that * X, E U , Ns( * X) is a condensing IMG space whose observable trace (nonstandard hull) Ns( * X)/E U is homeomorphic to X.
Let n be an arbitrary infinite hypernatural number and X = H ⊆ * X be the n-element hyperfinite set guaranteed by (iv) of 1.2.1. Now, it suffices to put X f = Ns( * X) ∩ X , E = E U ∩ (X × X), and we get another condensing IMG triplet (X, E, X f ) with the observable trace X f /E ∼ = Ns( * X)/E U ∼ = X and hyperfinite ambient space X .
The crucial property of the internal inclusion mapping Id X : X → * X (under the identification * x x x = x x x for x x x ∈ X) is namely that for each x x x ∈ X there is some x ∈ X such that x ≈ x x x. More generally, given any hyperfinite set X , an internal mapping η : X → * X is called a hyperfinite infinitesimal approximation, briefly an HFI approximation, of the Hausdorff uniform space X if: It is called an injective HFI approximation if, additionally, η is an injective mapping.
The standard counterpart of this notion can be formulated in terms of finite approximations and rasters. A raster in a uniform space (X, U), briefly an X-raster, is a pair (K, U) consisting of a compact set K ⊆ X and a symmetric entourage U ∈ U , such that U[x x x] ⊆ K for some x x x ∈ K. Given an X-raster (K, U), a mapping η : X → X is called a finite (K, U) approximation or just a (K, U) approximation of X if: If the map η is injective then it is called an injective (K, U) approximation.
A partially ordered set, briefly a poset (I, ≤), is a set I with a reflexive, antisymmetric and transitive binary relation ≤; if the relation ≤ is upward directed then (I, ≤) is called a directed poset. Obviously, the set of all X-rasters in a uniform space (X, U), partially ordered by the relation Let (X, U) be a Hausdorff uniform space and (I, ≤) be a directed poset. Then a system of mappings (η i : X i → X) i∈I is called an approximating system of the space (X, U) provided each X i is a finite set and, for any X-raster (K, U), there is i ∈ I such that η j : X j → X is a (K, U) approximation of X for each j ∈ I , j ≥ i.
It can be easily seen that every Hausdorff locally compact uniform space (X, U) has some approximating system (η i : X i → X) i∈I such that each X i is a finite subset of X and η i : X i → X is the inclusion mapping. Assuming κ-saturation for some sufficiently big κ, we can find an * X-raster K , U , formed by a * compact set K ⊆ * X and an entourage U ∈ * U , such that (K, U) ≤ (K , U ) for all X-rasters (K, U). Then there is i ∈ * I such that the hyperfinite set X i ⊆ * X (together with the inclusion map η i : X i → * X) is an internal K , U approximation of * X, hence an HFI approximation of X. Putting X = X i , E = E U ∩ (X × X), and X f = Ns( * X) ∩ X , we obtain a condensing IMG triplet (X, E, X f ) with a hyperfinite ambient space X and observable trace X = X f /E ∼ = X. This gives another proof of Proposition 1.2.2.

P Zlatoš
Given two IMG spaces (X, E, whenever x, y ∈ X f . In such a case we write f : (X, E, X f ) → (Y, F, Y f ). Note that every triplet morphism f with domain D can be formally extended to an everywhere defined triplet morphism f : X → Y in an arbitrary way.
Every triplet morphism f : for any internal functions f : This relation of infinitesimal nearness on finite elements is a monadic equivalence on any indicating a fundamental role of this indistinguishability equivalence. Let us remark, without being precise, that for a condensing IMG space (X, E, X f ) the equivalence ≈ Another important feature of condensing IMG spaces is the lifting property. Given a mapping f f f : X → Y between the observable traces, an internal mapping f : for each x ∈ X f . Notice that an internal mapping f : D → Y , satisfying the last equality is necessarily S-continuous on X f and satisfies f [ Then f f f necessarily is continuous, as well. Thus only continuous mappings between observable traces of IMG triplets have S-continuous liftings on X f . The point is that for a condensing IMG triplet (X, E, X f ) this necessary continuity condition is also sufficient for the existence of liftings.

Sketch of proof
Let's focus only on the nontrivial implication.
Assume that f f f is continuous and denote by Let us fix some nonempty internal set D ⊆ X f . Let further R, S be some systems of admissible size consisting of symmetric internal relations on X , Y respectively, such that E = R, F = S , and P ⊆ X f be a set of admissible size, such that R[P] = X f for every R ∈ R, whose existence is guaranteed in Proposition 1.2.1 (iii). Denote by I D the set of all ordered triples (S, R, A) such that S ∈ S , R ∈ R, A is a finite subset of P subject to D ⊆ R[A], and for all x, y ∈ D. Obviously, I D is of admissible size. Moreover, D ⊆ X is compact, hence f f f is uniformly continuous on D . Therefore, for any S ∈ S there is an R ∈ R and a finite A ⊆ P such that (S, R, A) ∈ I D . For any triple i = (S, R, A) ∈ I D we put: As A is finite, each Φ i is an internal relation. Using the continuity of f f f , the equality can be checked in a routine way, resembling the standard proof of the Closed Graph Theorem for a continuous function into a Hausdorff space.
E, X f ) satisfying the above condition will be called a triplet isomorphism. Since it is clear that, for any pair of triplet isomorphisms X form a pair of mutually inverse homeomorphisms, the last corollary can be briefly restated as follows: Condensing IMG triplets are isomorphic if and only if they have homeomorphic observable traces.
Corollary 1.2.5 Let X be a Hausdorff locally compact uniform space and (X, E, X f ) be a condensing IMG space with a hyperfinite ambient set X and the observable trace X homeomorphic to X. Then there is an HFI approximation η : X → * X which is a triplet isomorphism η : (X, E, X f ) → * X, E U , Ns( * X) .
In the last part of this section we briefly recall the Loeb measure construction and the construction of regular Borel measures on Hausdorff locally compact spaces by pushing down Loeb measures.
Let X = ∅ be a hyperfinite set and d : X → * R be an internal function such that d(x) ≥ 0 for each x ∈ X . Intuitively, d(x) is viewed as the "weight" of the point x.
Then d induces the internal (hyper)finitely additive measure ν d : P(X) → * R on the internal Boolean algebra P(X) of all internal subsets of X , given by we get a finitely additive (non-negative) measure • ν d : P(X) → R ∪ {∞} which has a unique extension to a σ -additive measure λ d : P(X) → R ∪ {∞} defined on the σ -algebra P(X) of all subsets of X generated by all monadic (or, equivalently, by all galactic) subsets of X . (If κ = ℵ 1 then P(X) is simply the σ -algebra generated by all internal subsets of X .) Then λ d is the Loeb measure induced by the internal function d (cf Cutland [7], Landers and Rogge [27], Loeb [28], or Ross [41]).
For the rest of this section, X denotes a Hausdorff locally compact space, represented as (and identified with) the observable trace X ∼ = X of a condensing IMG triplet (X, E, X f ) with a hyperfinite ambient set X by means of an HFI approximation η : X → * X.
Assume that the nonnegative internal function d : X → * R satisfies additionally the condition that ν d (A) is finite for each internal set A ⊆ X f . Then the set X f belongs to the algebra P(X) and the system Pushing down the Loeb measure along this map, ie putting for Borel Y ⊆ X, we obtain a regular Borel measure m m m d on X (which, if desirable, can be extended to a complete measure by the Carathéodory construction). The converse is essentially due to Anderson [3] who proved it in case m m m is a probability measure on X, which directly extends to the case that m m m(X) is finite. In the next proposition we manage without this assumption.
Sketch of proof There is a * X-raster (K, U) where U ∈ * U is * open in * X × * X and K ⊆ * X is a * compact set, such that U ⊆ E U , Ns( * X) ⊆ K, and η is a (K, U) approximation of * X. Since η has a hyperfinite range, there is an internal mapping σ : for x ∈ X . 1 The verification that the internal mapping d has all the required properties is left to the reader. Replacing each value d(x) by d(x) + ε, where ε is a positive infinitesimal such that ε |X| ≈ 0, we can satisfy also the additional requirement.

Bounded monadic groups
A bounded monadic group is an ordered triple (G, G 0 , G f ) consisting of an internal group G (which means that G is an internal set, endowed with internal operations of group multiplication and taking inverses), a monadic subgroup G 0 ⊆ G and a galactic subgroup Intuitively, G 0 is viewed as the subgroup of infinitesimals and G f is viewed as the subgroup of finite elements in G. Bounded monadic groups will be alternatively referred to as IMG group triplets (cf Ziman and Zlatoš [47]).
Every IMG group triplet gives rise to two IMG spaces: where E l , E r denote the left and the right equivalence relation on G corresponding to G 0 , respectively. Though they may differ and induce different uniformities on the ambient group G, they induce the same uniformity on G f ; in other words, the bounded monadic The reason is that the left and the right uniformities on G f are uniquely determined by the equivalence relations The group G f , as well as the observable trace G = G f /G 0 , endowed with the topologies induced by the subgroup G 0 , described in Section 1.2, become topological groups and the observable trace map x → x is a continuous surjective homomorphism of topological groups G f → G . (On the other hand, unless G 0 G, the topology on G induced by G 0 does not turn it into a topological group.) Two paradigmatic examples of IMG group triplets arise from the nonstandard extensions * R and * C of the additive groups of the fields R and C of the real and complex numbers, respectively. We denote by F * R ⊆ * R or F * C ⊆ * C the galactic subrings of finite (bounded) hyperreal or hypercomplex numbers, and by I * R ⊆ F * R or I * C ⊆ F * C the monadic ideals of infinitesimal hyperreal or hypercomplex numbers. Then ( * R, I * R, F * R) and ( * C, I * C, F * C) are IMG group triplets with observable traces F * R/I * R and F * C/I * C, isomorphic to the standard structures R and C, respectively, even as topological fields. For In any IMG group triplet (G, G 0 , G f ), the systems R and B from Section 1.2 can be replaced by a single system Q of admissible size, directed both upward and downward, consisting of symmetric internal subsets of G, such that: Pontryagin-van Kampen duality in the hyperfinite setting If G 0 is the intersection and G f is the union of countably many internal sets then we can assume that Q = {Q n : n ∈ Z} and the symmetric internal sets Q n ⊆ G satisfy: for all x, y, ∈ G ( (x) ≥ 0 already follows). Obviously, if is a gauge on G then x −1 y is a left invariant and xy −1 is a right invariant metric on G. Conversely, The set of all gauges on G is partially ordered by the relation A set V of gauges on G is called bidirected if it is both downward and upward directed with respect to the above partial order ≤.
Using a slightly modified Birkhoff-Kakutani style argument, one can prove the following version of the metrization theorem for bounded monadic group triplets.
If G 0 is an intersection and G f is a union of countably many internal sets, then there is a gauge : G → * R such that: In general, there is a bidirected set V of admissible size of gauges on G, such that: Sketch of proof In the countable case we give just the formula for ; the fact that it has all the required properties can be verified in a routine way.
There is a sequence (A n ) n∈Z of symmetric internal subsets of G such that for each n ∈ Z. By saturation, it can be extended to an internal sequence (A n ) −m≤n≤m for some m ∈ * N ∞ , such that A −m = {0}, A m = G, and the above inclusions hold for all n such that −m ≤ n < m. Let us put the internal set of all hyperfinite internal progressions in G. For any progression x = (x 1 , . . . , x n ) ∈ P(G) we denote by |x| = n its length and put: For the empty progression () = ∅ (ie, in case n = 0) this means that Π(∅) = 1 and w(∅) = 0. Then, finally, In the general case, let us invoke the system Q introduced in the beginning of this section. For each S ∈ Q there is a sequence A S n n∈Z of sets from Q such that A S 0 = S, for each n. Let S be the gauge constructed from (some hyperfinite extension of) this sequence. Let us denote: Then V 0 obviously has all the properties required for V , except perhaps for the bidirectedness. To fix this issue, we define for any gauges 1 , 2 and x ∈ G. As easily seen, both 1 ∧ 2 , 1 ∨ 2 are gauges, and: Taking for V the closure of V 0 with respect to the operations ∧ and ∨, we are done.
Remark The last metrization theorem can be proved even for more general triplets (G, G 0 , G f ), consisting of an internal group G, a monadic subgroup G 0 and a galactic subgroup G f , such that G 0 ⊆ G f ⊆ G, without the assumption that G 0 is normal in G f . Conversely, any single gauge , as well as any downward directed set V of admissible size of gauges on an internal group G gives rise to a monadic subgroup G 0 and a galactic subgroup G f , defined by the formulas from Proposition 1.3.1, such that G 0 ⊆ G f ⊆ G. However, one cannot prove that G 0 G f in general.
Topological groups G embeddable into observable traces G ∼ = G of IMG group triplets can be easily characterized.
Let G be a topological group and U be a symmetric neighborhood of the unit element 1 ∈ G. Then G is called U-locally uniform if the group multiplication in G restricted to the set U × U is uniformly continuous in the left (or, equivalently, in the right) uniformity on G (cf Goldbring [16]). G is locally uniform if it is U-locally uniform for some U.
Obviously, any subgroup of a locally uniform topological group is itself locally uniform (in the subgroup topology).
The easy proof of the following nonstandard formulation of U-local uniformity is left to the reader.
Since such sets U form a neighborhood base of 1 ∈ G, G is locally uniform. If G is just embedded into the observable trace G then G is isomorphic to a subgroup of a locally uniform group, hence it is locally uniform, as well. Now, assume that G is locally uniform. Let κ be the least uncountable cardinal, such that the topology of G has a base B of cardinality < κ, and * G be a nonstandard extension of G in a κ-saturated nonstandard universe. Let U ∈ B be a symmetric neighborhood of 1 ∈ G such that the group multiplication is uniformly continuous on U × U. Denote by I * G the monad of the unit element 1 ∈ * G and by N its normalizer in * G. Finally we put and denote by F * G = S the subgroup of * G generated by S. Then, obviously, F * G is a union of admissibly many internal sets, and * U ⊆ F * G ⊆ N, which means that I * G F * G. The inclusion G ⊆ F * G follows from the fact that, for each a a a ∈ G, the group operation is uniformly continuous on Ua a a × Ua a a, hence also on B × B where B ∈ B is a neighborhood of a a a such that B ⊆ Ua a a.
Thus ( * G, I * G, F * G) is an IMG group triplet and the star map a a a → * a a a induces an embedding of G into its observable trace * G = F * G/ I * G.
For locally compact groups even more can be proved. Proposition 1.3.4 Let G be a Hausdorff locally compact topological group. Then, in a sufficiently saturated nonstandard universe, G is isomorphic to the observable trace Ns( * G)/ Mon(1) of the condensing IMG group triplet * G, Mon(1), Ns( * G) .
Sketch of proof It suffices to have the nonstandard universe κ-saturated where κ is the least uncountable cardinal bigger than the cardinality of some neighborhood base of 1 ∈ G and such that G can be covered by the interiors of less than κ compact sets.
We are mainly interested in condensing IMG group triplets (G, G 0 , G f ) with a hyperfinite ambient group G. Condensing IMG triplets can be characterized using Proposition 1.
for any internal sets A, B between G 0 and G f .
Given a condensing IMG group triplet (G, G 0 , G f ) with a hyperfinite ambient group G, a positive number d ∈ * R, such that d |A| ∈ F * R I * R for some (or, equivalently, for each) internal set A between G 0 and G f , is called a normalizing multiplier or normalizing coefficient for (G, G 0 , G f ). According to 1.3.5, if d is a normalizing multiplier for (G, G 0 , G f ), and 0 < d ∈ * R then d is a normalizing multiplier if and only if d/d ∈ F * R I * R. In particular, for any internal set A between G 0 and G f , From the results of Section 1.2 it follows directly: If G is a locally compact topological group then any pair (K, U) consisting of a compact set K ⊆ G and a symmetric neighborhood U ⊆ K of the unit 1 ∈ G is called a G-raster. Notice that in such a case the neighborhood U is relatively compact, while P Zlatoš K itself is a compact neighborhood of 1 in G. The system of all G-rasters, partially ordered by the relation If G is discrete then it is enough to deal with its raster bases consisting just of pairs (K i , {1}), where K i are finite sets whose union is G. Similarly, if G is compact then it is enough to consider its raster bases formed by pairs (G, U i ) where the sets U i form a neighborhood base of 1 ∈ G.
Given a G-raster (K, U), a mapping η : G → G is called a finite (K, U) approximation or just a (K, U) approximation of G if G is a finite group, and η satisfies the following two conditions: for all x ∈ G, and the second of the above conditions can be strengthened to: However, the convenient conditions η(1) = 1, η x −1 = η(x) −1 alone can always be assumed without loss of generality.
Obviously, if (K, U) is a G-raster and η : G → G is a finite (K, U) approximation of G then η is a (Q, V) approximation of G for any G-raster (Q, V) ≤ (K, U).
If (I, ≤) is a directed poset then a system of mappings (η i : G i → G) i∈I , where each G i is a finite group, is called an approximating system of G if for every G-raster (K, U) there is i ∈ I such that η j : G j → G is a (K, U) approximation of G for each j ∈ I , j ≥ i.
An internal mapping η : G → * G is a hyperfinite infinitesimal approximation, briefly an HFI approximation, of G, if G is a hyperfinite group and: Pontryagin-van Kampen duality in the hyperfinite setting The notions of injective and strict HFI approximation, respectively, are defined in the obvious way.
Every hyperfinite infinitesimal approximation η : G → * G of a Hausdorff locally compact group G in a sufficiently saturated nonstandard universe gives rise to an IMG group triplet (G, G 0 , G f ), where Let G be a Hausdorff locally compact group. Then the following conditions are equivalent: (i) G is isomorphic to the observable trace G of some hyperfinite condensing IMG group triplet (G, G 0 , G f ). (ii) There is an HFI approximation η : G → * G of G.
(iii) For every G-raster (K, U) there is a finite (K, U) approximation η : G → G. (iv) There is an approximating system (η i : G i → G) i∈I of G by finite groups G i over some directed poset (I, ≤).
Proof (ii) ⇒ (i) was already proved, right before formulating the proposition.
(i) ⇒ (ii) Assume that (G, G 0 , G f ) is a hyperfinite condensing IMG group triplet and η η η : G f /G 0 → G is an isomorphism of topological groups. By Proposition 1.2.3, η η η has a lifting η : G → * G. It is a routine to check that η is an HFI approximation of G.  (iv) ⇒ (ii) Let (η i : G i → G) i∈I be an approximating system of G over (I, ≤). Let us embed the situation into a κ-saturated nonstandard universe, where κ > ℵ 0 and (I, ≤) has a cofinal subset of cardinality < κ. Then there is k ∈ * I such that i ≤ k for all i ∈ I ; it follows that η k : G k → * G is an HFI approximation of G.

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Remarks (a) From the above proof it is clear that one can strengthen conditions (ii)-(iv) of Proposition 1.3.7 by requiring injectivity of the corresponding approximation(s), and the modified conditions will be equivalent, again. What is not clear is the way one should modify condition (i) to make it equivalent with them. (Maybe even no modification is needed at all.) (b) Alternatively, the implication (iv) ⇒ (ii) can be proved using the ultraproduct construction with respect to any ultrafilter D on the directed poset (I, ≤) containing all the sets of the form {j ∈ I : j ≥ i} for i ∈ I , applied to the system of finite approximations (η i : G i → G) i∈I . More details can be found in Zlatoš [49, Proposition 1.5.8].
The following remark is even of more relevance.
Remark A notion of finite approximability equivalent, for compact metrizable groups, to the above condition (iii) (strengthened in the sense of Remark (a) above) was introduced and studied by Alan Turing [46] (3), admits such a representation (see also Gordon and Glebsky [12] and Gordon, Glebsky and Rubio [13]). Similarly, not even all finitely generated discrete groups are approximable by finite groups in this sense (cf Alekseev, Glebsky and Gordon [2], Gordon and Vershik [19]). Fortunately, as we shall see later on, all locally compact abelian groups still behave well.
The following examples (a), (b), (c), as well as some special cases of (d) are essentially taken from Gordon [18]; (b) in fact (without using the present terminology), can be found already in Luxemburg [32].  If n ∈ * N ∞ then the identity mapping Z n → * Z is a strict HFI approximation of Z. The IMG group triplet arising form it has the form Z n , {0}, Z .
(b) If n ∈ N and r > π/n then the homomorphism a → e 2πia/n : Z n → T is a strict injective (T, U) approximation of the group T, where U = {u ∈ T : |arg u| ≤ r}.
If n ∈ * N ∞ then the internal homomorphism a → e 2πia/n : Z n → * T is a strict injective HFI approximation of T. The corresponding IMG group triplet is (Z n , G 0 , Z n ), where: If n ∈ * N ∞ , and d is a positive infinitesimal such that nd ∼ ∞ then the internal mapping a → ad : Z n → * R is a strict injective HFI approximation of R, inducing the IMG group triplet (Z n , G 0 , G f ), where: Let G be any Hausdorff locally compact group (written multiplicatively) and U ⊆ K ⊆ G be its compact open subgroups such that U is normal in K. Then the quotient G = K/U is a finite group and any (necessarily injective) mapping η : G → G such that Uη(x) = x for each x ∈ G is an injective (K, U) approximation of G.
If G admits a raster base (K i , U i ) i∈I consisting of compact open subgroups U i K i of G then there are * compact * open subgroups U K of * G, such that U ⊆ Mon(1), Ns( * G) ⊆ K and the quotient G = K/U is a hyperfinite group. Any internal mapping η : G → * G such that Uη(x) = x for each x ∈ G is an injective HFI approximation of G. The IMG group triplet (G, G 0 , G f ) obtained from η satisfies: Both in the finite and in the hyperfinite case, η is not a strict approximation, unless it is a genuine homomorphism.

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Thus the items (a), (b) and (c) of the last example enable us to construct strict injective approximating systems and strict injective HFI approximations for all elementary LCA groups, ie for LCA groups of the form F × Z k × T m × R n where F is a finite abelian group and k, m, n ∈ N.
gives a direct hint how to construct both approximating systems and HFI approximations of any LCA group G with a raster base (K i , U i ) i∈I consisting of pairs of subgroups U i ⊆ K i of G. The following LCA groups are included as special cases: 1. the compact additive groups of τ -adic integers ∆ τ , where τ = (τ n ) n∈N is any increasing sequence of positive integers such that τ n | τ n+1 for each n 2. the additive LCA groups of τ -adic numbers Q τ , where τ = (τ n ) n∈Z is any sequence of positive integers, such that τ n < τ n+1 , τ n | τ n+1 for n ≥ 0, and τ n < τ n−1 , τ n | τ n−1 for n ≤ 0 3. torsion (discrete abelian) groups, as they are direct (inductive) limits of finite abelian groups 4. profinite (compact abelian) groups, ie, inverse (projective) limits of finite abelian groups 2 In Gordon [18] also HFI approximations of τ -adic solenoids which do not fall within the scope of Example 1.3.8(d), are described.
Actually, any LCA group admits arbitrarily good finite approximations and, henceforth, HFI approximations, too. The key ingredient of the proof of this fact is the following structure theorem for compactly generated LCA groups (ie, LCA groups generated by some compact subset  Equivalently, G admits some approximating system (η i : G i → G) i∈I by finite abelian groups, such that all the mappings η i are injective and preserve 0 and inverses.
Proof Let us pick a compact symmetric neighborhood V of 0 such that V + V ⊆ U. According to the above Proposition 1.3.10, V contains a closed subgroup H such that the quotient E = K /H of the subgroup K of G generated by the compact set K is an elementary LCA group. Denote by V and K the images of the sets V, K, respectively, under the canonical projection ψ : K → E. Then V is a symmetric neighborhood of 0 and K is a compact set in E. According to Example 1.3.8 and the comment following it, there is a finite group G and a strict injective K , V approximation ζ : G → E. Let σ : E → K be any (necessarily injective) mapping such that σ(0) = 0, σ(−x) = −σ(x) and ψ(σ(x)) = x for x ∈ E. Then η = σ • ζ : G → G is an injective mapping, satisfying η(0) = 0, and η(−a) = −η(a) for a ∈ G. Straightforward arguments show that for a, b ∈ G, whenever η(a), η(b), η(a+b) ∈ K+H. Hence, η is a (K, U) approximation of G. The equivalence of the first and the second formulation is obvious in view of Proposition 1.3.7. Corollary 1.3.11 (The Hyperfinite LCA Group Approximation Theorem) Let G be a Hausdorff LCA group. Then, in a sufficiently saturated nonstandard universe, there is an internal hyperfinite abelian group G and an injective HFI approximation η : G → * G of G, such that η(0) = 0 and η(−a) = −η(a) for a ∈ G. It follows that G is isomorphic to the observable trace G = G f /G 0 of the IMG group triplet (G, G 0 , G f ) with a hyperfinite abelian ambient group G, where G 0 = η −1 Mon(0) and G f = η −1 Ns( * G) .

Pontryagin-van Kampen duality in the hyperfinite setting
In this chapter we finally come to the central topic of our paper, which is the study of condensing IMG group triplets with a hyperfinite abelian ambient group and their relation to the Pontryagin-van Kampen Duality Theorem. In particular, we will prove the first and the second of the three Gordon's conjectures concerning them. However,

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we start with a technical section dealing mainly with the additive combinatorial structure of some sets in finite abelian groups.

Fourier transforms, Bohr sets and spectral sets in finite abelian groups
In this section we establish some inclusions between certain types of subsets of finite abelian groups and their dual groups, as well some estimates of their size. Later on, we will apply them within the realm of hyperfinite abelian groups by means of the Transfer Principle. To this end we will systematically employ the discrete Fourier transform as our main tool. The results thus obtained will not depend on the normalizing coefficients occurring in it. As a matter of convenience we choose d = 1/|G| and d = 1.
In what follows G denotes a finite abelian group with the addition operation. The operation on its dual group G is written as the multiplication. The set C G of all functions G → C becomes a unitary space endowed with the Hermitian inner product, given as the expectation for f , g ∈ C G , and the corresponding L 2 -norm: The dual group G = Hom(G, T), considered as a subset of C G , forms an orthonormal basis of C G . Thus for the Fourier transform F : the Fourier inversion formula takes the form: The Hermitian inner product on C G ensures the Plancherel identity: The convolution on C G is also defined as the expectation: Journal of Logic & Analysis 13:1 (2021) Pontryagin-van Kampen duality in the hyperfinite setting

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For its Fourier transform we have: Additionally, we will make use of the L p -norms on C G and the p -norms on C G Let us list some well known and/or obvious relations:

are special cases of the Hausdorff-Young inequality
f q ≤ f p for 1 ≤ p ≤ 2 and q being the dual exponent of p, ie, 1/p + 1/q = 1.
We denote by f a the shift of the function f : G → C by the element a ∈ G, ie for γ ∈ G, as well as A norm N on the linear space C G is called translation invariant if N( f a ) = N( f ) for all f ∈ C G , a ∈ G. For any translation invariant norm N we have: For any absolute norm N we have: As all the norms · p are obviously translation invariant and absolute, If f , g : G → R are both nonnegative on G then even: For the characteristic function (indicator) 1 A of a set A ⊆ G and 1 ≤ p < ∞ we have: then so is f : G → C and we have: The support of a function g : G → C is defined as the set: The following estimate generalizes an inequality in Green and Ruzsa [20], which was part of the proof there of Proposition 3.1, from indicators to arbitrary nonnegative functions.
Lemma 2.1.1 Let f : G → R be nonnegative, and D ⊆ G be a nonempty set such that supp( f * f ) ⊆ D. Then: Proof By the Plancherel identity and the relation between the Fourier transform and the convolution, for any function f : G → C. Using Cauchy-Schwartz inequality, the fact that f * f is supported on D, and the nonnegativity of f , we get: The claim follows immediately. For any subsets A ⊆ G, Γ ⊆ G and any α ∈ * R, 0 ≤ α ≤ π , we introduce the Bohr sets (cf Tao and Vu [44]): The spectral set or the spectrum of a function f : G → C at the threshold t ∈ R is the set: This is a slight generalization of a definition from Tao and Vu [44, § 4.6] were spectral sets of subsets A ⊆ G were defined by: and f is not identically 0; similarly, Spec t ( f ) = G for t ≤ 0. Thus it makes sense to consider the spectral sets just for the threshold values 0 ≤ t ≤ 1. Also the following implication is trivial:

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Proof Take any γ ∈ Bohr α (D). According to the assumptions on f , t and α we have The following result generalizes an inclusion proved in Green and Ruzsa [20] during the proof of Proposition 3.1, as well.
Proposition 2.1.3 Let f : G → R be a nonnegative function, not identically equal to 0, D ⊆ G be a nonempty set such that supp( f * f ) ⊆ D, and 0 < α < π/2, 0 ≤ t ≤ 1. Then we have Re γ(x) ≥ cos α for γ ∈ Γ . By the Fourier inversion formula, Lemma 2.1.1 and Plancherel identity we get: Pontryagin-van Kampen duality in the hyperfinite setting The strict inequality in the third line is due to the fact that for the trivial character 1 G ∈ Γ we have Re 1 G (x) = 1 > cos α, since α > 0. According to the assumption on t, the expression in the last line is ≥ 0.
Remarks (a) It would be enough to have the above result for a single fixed value α.
The authors both in Green and Ruzsa [20] as well as in Tao and Vu [44] take α = π/3, which, in our case, would result in the estimate: It is worthwhile to notice that in the special case of f being the indicator of a nonempty set A ⊆ G and D = A + A we have where σ(A) = |A + A| / |A| is the doubling constant of A.
We will need also some lower and upper bounds of the size of spectral sets of some functions.
Proposition 2.1.4 Let f : G → R be a nonnegative function, not identically equal to 0, D ⊆ G be a nonempty set such that supp( f * f ) ⊆ D, and 0 < t ≤ 1. Then: Proof Let us denote Γ = Spec t ( f ). According to Lemma 2.1.1 we have using Plancherel formula to pass to the last line. This gives the lower bound.

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The upper bound readily follows from the following computation: Remark Notice that the lower bound is relevant just in case otherwise it is trivial. One of its consequences can be stated as in which form it can be regarded as a kind of the Uncertainty Principle, setting some lower bound for the expression on the left. For f = 1 A being the indicator of a nonempty set A ⊆ G the first inequality gives Let us close this technical section with a kind of the Smoothness-and-Decay Principle indicating that spectral sets of functions continuous in some sense tend to avoid discontinuous characters. We will devote a whole section to this topic in our next paper [50].
Proposition 2.1.5 Let f : G → C be not identically equal to 0 and C, D ⊆ G satisfy supp f ∪ (supp f + C) ⊆ D = ∅. Let 0 < t ≤ 1, 0 < α < π and ε > 0 be real numbers. Assume that Now, take any γ ∈ Spec t ( f ) and assume that γ / ∈ Bohr α (C). Then there is a ∈ C such that |γ(a) − 1| > 2 sin(α/2). Thus we have A brief inspection of the proof yields the following modification of the last result.

The dual triplet
Throughout this and the next section, (G, G 0 , G f ) denotes a fixed but arbitrary condensing IMG group triplet with a hyperfinite abelian ambient group G (tacitly assumed to be externally infinite). 3 We also fix a system Q of admissible size, directed both downward and upward, consisting of symmetric internal sets such that G 0 = Q and G f = Q.
Let us denote by G = G = G f /G 0 the observable trace of the triplet. The LCA group G gives rise to the dual group G of all continuous homomorphisms (characters) γ γ γ : G → T, endowed with the compact-open topology. Then G, as an LCA group, can itself be represented as the observable trace of some condensing IMG group triplet (H, H 0 , H f ) with a hyperfinite abelian ambient group H . One can naturally expect that at least one from among the triplets representing the dual group G can be constructed from the original triplet (G, G 0 , G f ) in some canonical way.
Let G = Hom(G, * T) denote the set of all internal homomorphisms (characters) γ : G → * T; then G with the pointwise multiplication is a hyperfinite abelian group internally isomorphic to G (though not in a canonical way). For any sets A ⊆ G, Γ ⊆ G we define their infinitesimal annihilators by: Obviously, A is a subgroup of G and Γ is a subgroup of G. For any A, B ⊆ G, 3 However, the reader should keep in mind that some of our accounts remain valid for general internal abelian ambient group G, as well, or require just some minor modification.

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showing that the assignments A → A , Γ → Γ form a Galois connection.
We are particularly interested in the subgroups of the dual group G. As every γ ∈ G is a group homomorphism, it belongs to G 0 if and only if it is S-continuous as a mapping γ : G → * T with respect to the monadic equivalence E l = E r on G and the usual equivalence of infinitesimal nearness ≈ on * T, inherited from the hypercomplex plane * C. The elements of G 0 play the role of finite or accessible characters. The characters γ ∈ G f are infinitesimally close to 1 on the whole subgroup G f of finite elements of G, ie, in front of the horizon they are indistinguishable from the trivial character 1 G ∈ G. They play the role of infinitesimal characters. However, this intuition calls for some justification.
For A ⊆ G and any subset T of the interval (0, π] ⊆ R, such that inf T = 0, we obviously have: The point is that for a subgroup A of G a single α suffices. Proof As each γ ∈ G is a homomorphism, for a subgroup A ⊆ G and γ ∈ Bohr α (A), the image γ[A] must be a subgroup of * T contained in the arc {c ∈ * T : |arg c| ≤ α}. For α < 2π/3, however, the biggest subgroup of * T contained there is namely the while the reversed inclusion is trivial.
If A is an internal subgroup of G and 0 < α < 2π/3 then a similar argument shows that both the sets coincide with the (strict) annihilator A ⊥ of A; more precisely: Proof Let us pick any (standard) α ∈ (0, 2π/3). As G 0 , G f are subgroups of G, according to the last lemma we have: While the last equation in the second line is trivial, the last equation in the first one follows by a straightforward saturation argument. As all the sets Bohr α (Q) are internal, G 0 is a galactic set and G f is a monadic one. It remains to show that the IMG group To this end consider any symmetric internal sets Γ , ∆ such that G f ⊆ Γ ⊆ ∆ ⊆ G 0 . Then there are P, Q ∈ Q such that G 0 ⊆ P ⊆ Q ⊆ G f and: Using an elementary combinatorial argument we will show that ∆ : Γ ≤ Bohr α/4 (P) : where the upper integer part 4π/α equals the covering index T : S of the circle T with respect to the arc S = {c ∈ T : |arg c| ≤ α/4}. Indeed, let k = T : S and q = Q : P . Then there are k points c 1 , . . . , c k ∈ T and q points x 1 , . . . x q ∈ G such that T ⊆ k i=1 Sc i and Q ⊆ q j=1 P+x i . Let h : T → {c 1 , . . . , c k } be any function such that c ∈ S h(c) for c ∈ T. As there are only k q functions {x 1 , . . . , x q } → {c 1 , . . . , c k }, given more then k q functions in Bohr α/4 (P), there are at least two, γ and χ, say, such Let's choose any x ∈ Q and j ≤ q such that x ∈ P + x j . Then γ(x)γ(x j ) −1 , χ(x)χ(x j ) −1 ∈ S and, as h(γ(x j )) = h(χ(x j )), we have: ie γχ −1 ∈ Bohr α (Q). It follows that Bohr α/4 (P) : Bohr α (Q) ≤ k q .
The condensing IMG group triplet G, G f , G 0 will be called the dual triplet of the IMG group triplet (G, G 0 , G f ).

P Zlatoš
For an internal character γ ∈ G 0 there are potentially two interpretations of its observable trace γ . First, it is simply the element G f γ of the quotient G 0 /G f = G . Second, γ is the observable trace of the S-continuous mapping γ : G → * T, ie for x ∈ G f . That way γ : G → T is a continuous character of the LCA group G (cf Section 1.3). The assignment γ → γ , depicted in the commutative diagram justifies the identification of the "two observable traces" G f γ and γ . As proved by Gordon in [17] (see also [18]), even more is true.
where Q ∈ Q, form a neighborhood base of the unit character in G. It follows that the As a continuous image of an LCA (hence complete) topological group, it is necessarily closed.
It is both natural and tempting to conjecture that the canonical mapping G → G is also surjective, ie that it is an isomorphism of topological groups. This is indeed the first of Gordon's Conjectures from Gordon [17] (see also [18, page 132]). Sketch of proof G obviously separates points in its dual group G which, according to the Duality Theorem, can be identified with G. Now, it can be easily seen that any dense subgroup D ⊆ G separates points in G, as well.
Conversely, if D is not dense in G then its closure D is a proper closed subgroup of G. As a consequence of the Duality Theorem (cf Morris [36, Corollary 1 to Theorem 21]), there is a nontrivial character of G, ie, an element x x x ∈ G {0} such that γ γ γ(x x x) = 1 for all γ γ γ ∈ D. Therefore, neither D nor D separate points in G.
Pontryagin-van Kampen duality in the hyperfinite setting 45 The following example shows that (a) the condition X = X is not automatically satisfied for monadic or galactic subgroups X of hyperfinite abelian groups, and (b) there are plenty of subgroups of hyperfinite abelian groups satisfying this condition even without being monadic or galactic.
Example 2.2.8 An additive cut on * N is any nonempty subset C ⊆ * N, such that: Assume that C is an additive cut on * N, N ⊆ C, and n ∈ * N C.
(a) Let Z be any nontrivial finite abelian group, eg Z = Z d for some 2 ≤ d ∈ N. Let G = Z n be the hyperfinite abelian group of all internal sequences z = (z 1 , . . . , z n ) of elements from Z . Denote for z ∈ G, and put: S(C) = {z ∈ G : |supp z| ∈ C} As C = {0} and n / ∈ C, as well as supp(−z) = supp z and supp(y+z) ⊆ supp y∪supp z for any y, z ∈ G, S(C) is a nontrivial proper subgroup of G. Moreover, if C is a monadic (galactic) set then so is S(C). On the other hand, all the internal subgroups where 1 ≤ k ≤ n, satisfy G k ⊆ S(C). Identifying the internal dual G with the hyperfinite abelian group Z n in the obvious way, each character γ ∈ G is represented as the ordered n-tuple γ = (γ 1 , . . . , γ n ), where γ k ∈ Z . Then as d |D| < ∞. Now, for any γ ∈ G G 0 there is some a ∈ G 0 such that γ(a) ≈ 1.
As γ(a) − 1 ≈ 0, the conclusion f (γ) ≈ 0 follows immediately. Then Remark Both Proposition 2.3.1 and its Corollary 2.3.2, are "soft" statements in the sense of Tao's blog which appeared in his book [43, §1.3, §1.5]. Their "hard" version is Proposition 2.1.5 and its Corollary 2.1.6, providing us with the additional information how one has to choose ε > 0 and the internal set C, such that G 0 ⊆ C ⊆ G f , in order to On the other hand, the last corollary is fully sufficient for our purpose. We hope that at least some readers will appreciate the advantage of nonstandard arguments, allowing one to dispense with lots of meticulous estimates and "epsilontics".
As made clear in Section 2.2, in order to finish the proof of the Triplet Duality Theorem 2.2.5, as well as of Gordon's Conjecture 1 (Theorem 2.2.4), it is enough to prove the following: Proposition 2.3.3 Let (G, G 0 , G f ) be a condensing IMG group triplet with hyperfinite abelian ambient group G. Then: is obvious, it suffices to prove the opposite inclusion. Let V be a set of internal gauges on G such that G 0 = {x ∈ G : (∀ ∈ V)( (x) ≈ 0)} whose existence was established in Proposition 1.3.1. For any ∈ V and a positive r ∈ * R we denote by B (r) = {x ∈ G : (x) ≤ r} the internal closed ball of radius r, and define the internal function h r : G → * R by: Obviously, h r is even, nonnegative, and supp h r ⊆ B (r). For 0 ≈ r < ∞, both the sets supp h r and B (r) are between G 0 and G f . Moreover, for r noninfinitesimal, h r is also S-continuous and h r 1 ≥ d |B (r/2)|/2 ≈ 0. By Corollary 2.3.2, Thus picking some standard positive α < π/2, there is a standard t > 0 with: For such α and t we have: Consequently: As and r were arbitrary, this is enough to establish the inclusion G 0 ⊆ G 0 .
Remark In fact the subgroup G f plays no role in the above statement nor in its proof. The only thing we need to assume is the existence of an internal set A subject to G 0 ⊆ A ⊆ G such that |A|/|B| < ∞ for each internal B such that G 0 ⊆ B ⊆ A.
Having d as a normalizing multiplier for the triplet (G, G 0 , G f ), then, in order to get the Fourier inversion formula and Plancherel identity, we need to normalize the inner product, the Fourier transform, etc, on the space * C G of internal functions G → * C, defined on the internal dual group G, by means of the normalizing coefficient: In view of the canonical isomorphism of the observable trace G = G 0 /G f and the dual group G , there naturally arises the question whether such a d is a normalizing multiplier for the dual triplet G, G f , G 0 . In that case (and only in that case) the measure m m m d , obtained by pushing down the Loeb measure λ d from G to the observable trace G ∼ = G , will be a Haar measure on G . The second of Gordon's conjectures, whose proof is the second main result of this paper, states that the response to the above question is affirmative. Theorem 2.3.4 (Gordon's Conjecture 2) Let (G, G 0 , G f ) be a condensing IMG group triplet with a hyperfinite abelian ambient group G. Then, for any internal set D such that a normalizing multiplier for (G, G 0 , G f ) and d = |D|/|G| is a normalizing multiplier for the dual triplet G, G f , G 0 . More generally, if d is any normalizing multiplier for (G, Proof Since, for any internal set D such that G 0 ⊆ D ⊆ G f , d = 1/ |D| is a normalizing multiplier for the triplet (G, G 0 , G f ) and d 1 /d 2 ∈ F * R I * R for any its normalizing multipliers d 1 , d 2 , it suffices to find a single couple of internal sets D between G 0 , G f and Γ between G f , G 0 , such that the quotient |D| |Γ | /|G| is neither infinite nor infinitesimal.
Starting with any internal gauge ∈ V , we have G 0 ⊆ B (r) ⊆ G f whenever 0 < r ∈ R. Also, we can take arbitrary standard α and t subject to 0 < α < π/2, 0 < t ≤ cos α. Then the spectral set Γ = Spec t (h r ) is internal and, by Lemma 2.1.
the upper bound in the preceding inequality is finite for any standard t > 0. For the same reason, it is possible to find a standard t, such that 0 < t ≤ cos α, and making the lower bound positive and noninfinitesimal.

Some (mainly) standard equivalents: Hrushovski style theorems
In this section we list some direct standard consequences (in fact, equivalents) of way of argumentation, we give just the proof of the former (which is technically more complicated), introducing an additional nonstandard equivalent formulated as Theorem 2.4.1.
As we will show within short, Gordon's Conjecture 1 (Theorem 2.2.4) is equivalent to a kind of almost-near or stability principle for characters of hyperfinite abelian groups.
The following notions are needed for the nonstandard formulation of this principle.
Let G be an internal abelian group and X be a (not necessarily internal) subset of G.
Two internal mappings f : C → * T, g : D → * T, defined on sets C, D ⊆ G, are said to be infinitesimally close on a set for all x ∈ X . We say that g is almost homomorphic on a set X if X ⊆ D and, for all x, y ∈ X , x + y ∈ X implies: Theorem 2.4.1 Let (G, G 0 , G f ) be a condensing IMG group triplet with a hyperfinite abelian ambient group G, and g : D → * T be an internal mapping, where G f ⊆ D ⊆ G.
If g is S-continuous and almost homomorphic on G f then there exists an internal S-continuous character γ : G → * T, ie γ ∈ G 0 , such that g and γ are infinitesimally close on G f . Due to saturation, any internal mapping g : D → * T, where G f ⊆ D ⊆ G, which is almost homomorphic on G f , must be almost homomorphic on some symmetric internal set C such that G f ⊆ C ⊆ D. Similarly, if g and γ are infinitesimally close on G f then they must be infinitesimally close on some symmetric internal set between G f and D. At the same time, internal mappings g : D → * T almost homomorphic on the galaxy G f share the following property with internal homomorphisms γ ∈ G: such a g is S-continuous on G f if and only if, given any (standard) α ∈ (0, 2π/3), there is some internal set A between G 0 and G f such that |arg g(x)| ≤ α for all x ∈ A (cf Lemma 2.2.1 and the beginning of the proof of Proposition 2.2.2).
The reformulation of Theorem 2.4.1 in standard terms is a highly uniform, but rather cumbersome stability principle for characters of finite abelian groups. For the sake of its formulation, as well as of the proof of the equivalence of Theorems 2. Let G be an abelian group, and ε > 0 be a (standard) real. Two mappings f : C → T, g : D → T, defined on subsets C, D ⊆ G, are said to be ε-close on the set X ⊆ C ∩ D if for all x ∈ X . We say that g is ε-homomorphic on the set X ⊆ D if for all x, y ∈ X the condition x + y ∈ X implies: In Theorem 2.4.2 Let α, ε ∈ (0, 2π/3), k ≥ 1 and (q j ) ∞ j=1 be any sequence of reals q j ≥ 1. Then there exist m ≥ 1, n ≥ k and δ > 0, depending just on α, ε, k and the sequence (q j ), such that the following holds: Let G be a finite abelian group and 0 ∈ A n ⊆ . . .
Then, for every partial δ -homomorphism g : mA 0 → T such that |arg g(x)| ≤ α for x ∈ A k , there exists a homomorphism γ : G → T such that g and γ are ε-close on A 0 . Then we can fix some α, ε, k and a sequence (q j ) witnessing a counterexample. Let (δ n ) be any strictly decreasing sequence such that δ n → 0. Then for all m = n ≥ k there exist a finite abelian group G n and symmetric sets 0 ∈ A nn ⊆ . . . ⊆ A n1 ⊆ A n0 ⊆ G n such that A nj + A nj ⊆ A nj−1 and [A nj−1 : A nj ] ≤ q j for 1 ≤ j ≤ n, as well as a partial δ n -homomorphism g n : nA n0 → T such that |arg g n (x)| ≤ α for x ∈ A nk . On the other hand, for every genuine homomorphism γ n : G n → T, there is x n ∈ A n0 such that: Let D be any nonprincipal ultrafilter on the set I = {n ∈ N : n ≥ k}. Then the ultraproduct G = n∈I G n /D of the finite groups G n is a hyperfinite abelian group (within the ℵ 1 -saturated nonstandard universe obtained via the ultraproduct construction modulo D ). For each n ∈ I , n < j, we put A nj = {0} and take the ultraproduct A j = n∈I A nj /D considered as an internal subset of G. Finally, we put: Then G 0 , as an intersection of countably many internal sets, is a monadic subgroup of G and G f , as a union of countably many internal sets, is a galactic subgroup of G.
Obviously, G 0 ⊆ G f . By Łos' Theorem, [A j−1 : A j ] ≤ q j , as well as [jA 0 : A 1 ] ≤ q j 1 for each j ≥ 1. It follows easily that [V : U] < ∞ for any internal sets U , V such that G 0 ⊆ U ⊆ V ⊆ G f , hence, by Proposition 1.3.5, (G, G 0 , G f ) is a condensing IMG triplet with a hyperfinite abelian ambient group G.
Let us form the internal mapping g = (g n ) n∈I /D . Then, for each n, g is a partial δ n -homomorphisms from the internal set n∈I nA n0 /D ⊇ G f to the ultrapower group * T = T I /D . Thus g is almost homomorphic on G f and it maps the internal set A k = n A nk /D , satisfying G 0 ⊆ A k ⊆ G f , into the arc {c ∈ * T : |arg c| ≤ α}. It follows that g is S-continuous on G f . Then the observable trace g = • (g G f ) is a continuous character of the LCA group G = G f /G 0 . By Theorem 2.2.4, there is an On the other hand, there is a set J ∈ D such that γ n ∈ G n for all n ∈ J . By our assumptions, for each n ∈ J , there is x n ∈ A n0 such that arg γ n (x n )/g n (x n ) > ε. Let x n = 0 for n ∈ I J . Then, for Remark Concerning m, n the result is purely existential, giving no upper bound for them (though we can always have m = n). On the other hand, it seems rather probable that even a more uniform version of Theorem 2.4.2 is true. We conjecture that, similar to Mačaj and Zlatoš [33,Theorem 4.1], one can take any δ > 0, such that δ < min{ε, π/2, 2π/3 − α}, and choose m, n depending additionally on δ .
Proof of 2.4.2 ⇒ 2.4.1 Let (G, G 0 , G f ) be a condensing IMG group triplet with a hyperfinite abelian ambient group G, D be an internal set such that G f ⊆ D ⊆ G and g : D → * T be an internal mapping that is S-continuous and almost homomorphic on G f . We can also assume that g(0) = 1. Thus fixing a standard α ∈ (0, 2π/3), there is a symmetric internal set V between G 0 and G f such that |arg g(x)| ≤ α for x ∈ V . Let us choose, additionally, a β ∈ (α, 2π/3).
It suffices to show that, for each symmetric internal set A such that V + V ⊆ A ⊆ G f , and each standard ε > 0, such that α + ε ≤ β , there is γ ∈ Bohr β (V) such that for all x ∈ A. Then, due to saturation, there is γ ∈ Bohr β (V) ⊆ G 0 such that γ(x) ≈ g(x) for all x ∈ G f , ie, γ and g are infinitesimally close on G f . So let us fix some A, V , α, β and ε, satisfying the above assumptions. There is a sequence of symmetric internal sets (A j ) j∈N such that A 0 = A, A 1 = V , A j+1 +A j+1 ⊆ A j and G 0 ⊆ A j for each j. Put q j = [A j−1 : A j ] for 1 ≤ j ∈ N. Let m, n and δ be the numbers guaranteed for α, ε, k = 1 and the sequence (q j ) by Theorem 2.4.2. By the Transfer Principle they have to work for the hyperfinite abelian group G and the internal sets A 0 , . . . , A n as well. Since g is almost homomorphic on G f , it is δ -homomorphic on mA 0 ⊆ G f . It follows that there is an internal character γ ∈ G such that arg g( Let us close the cycle by proving that Theorem 2.4.1 implies Gordon's Conjecture 1 (Theorem 2.2.4), ie the surjectivity of the canonical embedding G → G . In view of Proposition 1.2.3, the proof of this implication is plain.
Proof of 2.4.1 ⇒ 2.2.4 Let (G, G 0 , G f ) be a condensing IMG group triplet with a hyperfinite abelian ambient group G and G = G f /G 0 denote its observable trace. Assume that γ γ γ : G → T is a continuous character of G. By Proposition 1.2.3, there is an internal mapping g : D → * T such that G f ⊆ D ⊆ G, g is S-continuous on G f , and for all x ∈ G f . Then g(x + y) ≈ γ γ γ (x + y) = γ γ γ x + y = γ γ γ x γ γ γ y ≈ g(x) g(y) for all x, y ∈ G f , ie, g is almost homomorphic on G f . By 2.4.1, there is a γ ∈ G 0 such that γ(x) ≈ g(x) ≈ γ γ γ x for each x ∈ G f , ie, γ = γ γ γ .
A brief inspection of the above proofs is instructive. In order to derive Theorem 2.4.2 from Theorem 2.2.4 it suffices to assume that Gordon's Conjecture 1 is true just for condensing IMG group triplets (G, G 0 , G f ) with a hyperfinite G (in some ℵ 1 -saturated nonstandard universe), such that G 0 is an intersection and G f is a union of countably many internal sets. This was indeed Gordon's original formulation of his conjecture in [17]. Theorem 2.4.3 Let α, β ∈ (0, 2π/3) and (q j ) ∞ j=1 be any sequence of reals q j ≥ 1. Then there exists n ∈ N, depending just on α, β and the sequence (q j ), such that the following holds: Let G be a finite abelian group and 0 ∈ A n ⊆ . . . ⊆ A 1 ⊆ A 0 ⊆ G be symmetric sets such that 2) remain true when replacing "Let G be a finite abelian group and 0 ∈ A n ⊆ . . . ⊆ A 1 ⊆ A 0 ⊆ G be symmetric sets. . . " by "Let G be a Hausdorff LCA group and A n ⊆ . . . ⊆ A 1 ⊆ A 0 be compact symmetric neighborhoods of 0 in G. . . " (without changing the rest) in their formulation.
In order to prove all that, it would be necessary (and sufficient) to deal with condensing IMG triplets of the form (G, G 0 , G f ), where G is an arbitrary internal Hausdorff LCA group, ie, a * (Hausdorff LCA) group, and define their dual triplets G, G f , G 0 , with G denoting the internal dual group of G, as well as to generalize the results of Section 2.1 from finite abelian groups to arbitrary Hausdorff LCA groups. That, however, though possible and -as we shall see in the next section -unavoidable to some extent, would be at odds with the leading intentions of the present paper, namely to study the LCA groups and the Fourier transform on functional spaces over them by means of their approximations by (hyper)finite abelian groups and the discrete Fourier transform on them.
Analogous remarks could be made on behalf of the indicated generalizations of Theorems 2.4.1 and 2.4.2 and their relation to some standardly formulated stability results for characters of LCA groups with respect to the compact-open topology from Mačaj and Zlatoš [33] and Zlatoš [48]. However, the big amount of technicalities needed for their formulation and comparison would take us too far from the main lines and goals of the present paper.

Simultaneous approximation of an LCA group and its dual
Now, we are going to apply the results of the previous sections to triplets arising from HFI approximations of LCA groups. One can expect that an HFI approximation η : G → G of an LCA group G gives rise to an HFI approximation φ : G → G of the dual group G. This is true in some sense, however, in general, the connection between these two approximations is not as straightforward as one wished. Some of the material of this section is partly covered in the Introduction and § 2.4 of Gordon's book [18].
As we will be dealing with plenty of more or less canonical isomorphisms of topological groups, first we have to make clear which of them we intend to exploit for identification of the isomorphic objects, and which of them we still view as isomorphisms of different objects.
Let G be a Hausdorff LCA group, viewed as a subgroup of its nonstandard extension * G. Let us denote: and Then * G, I * G, F * G is a condensing IMG group triplet with an internal abelian ambient group * G. We identify the canonically isomorphic LCA groups G and the observable trace F * G/I * G (nonstandard hull) of this triplet, as well as the standard part map x x x → • x x x : F * G → G and the observable trace map x x x → x x x : F * G → F * G/I * G.
Similar, but less straightforward, identification applies also to the dual group G. More specifically, due to the Transfer Principle, the nonstandard extension * G of the dual group G coincides with the internal dual group * G of * G, consisting of all internal * continuous characters γ γ γ : * G → * T. However, the elementary embedding G → * G Proof (a) In order to prove the first inclusion, take any γ ∈ H f . As The third and the forth inclusion now follow by the symmetry of the situation.
(b) For brevity's sake let us number the inclusions consecutively from (i) to (iv). First (ii) ⇒ (iii) can be proved analogously by a symmetry argument.
Next we show (iv) ⇒ (ii). From G f ⊆ H 0 we get H 0 ⊆ G f = G f by the Triplet Duality Theorem 2.2.5 (see also Lemma 2.2.7). By symmetry we have (iii) ⇒ (i) as well, closing the circle of implications.
In other words, under the assumptions of the lemma, the HFI approximations η and φ form an adjoint pair if and only if any (hence all) of the inclusions from (b) is true. We are inclining to consider the first inclusion G 0 ⊆ H f from (b) as intuitively the most appealing and fundamental one. In "unzipped form" it states that, for all γ ∈ G, and, as γ(x) ≈ (φ γ)(η x), it is equivalent to: The remaining inclusions in (b) can be "unzipped" in a similar way.
The standard meaning of the above accounts and results can be formulated in terms of adjoint pairs of finite approximations. The equivalence of both formulations could be established by referring to Nelson's translation algorithm [37]. However, this would obscure some connections, so we offer a more detailed exposition.
A (K, U) approximation η : G → G and a (Γ Γ Γ, Ω Ω Ω) approximation φ : G → G are called pairing preserving on the set for any An α-adjoint approximation scheme is called pairing preserving if the approximations η , φ are pairing preserving on K × Γ Γ Γ.
The following inclusions form the standard analogue of part (a) of Lemma 2.5.3.
Lemma 2.5.4 Let G be a Hausdorff LCA group, G be a finite abelian group and α > 0, ε ≥ 0 be such that α + ε < 2π/3. If (K, U), (Γ Γ Γ, Ω Ω Ω) are rasters and η : G → G, φ : G → G are mappings forming an α-adjoint ε-pairing preserving approximation scheme of the pair G, G then Proof We will prove just the upper left inclusion; the lower left inclusion is equivalent to it, while the inclusions in the right column follow from those in the left one by a symmetry argument. We have to show that |arg γ(x)| ≤ α + ε for any In that case we have In order to take care of (the first two of) the four equivalent conditions formulated in part (b) of Lemma 2.5.3, we need to introduce one more notion. Assume that (Q, V) is a G-raster and (∆ ∆ ∆, Υ Υ Υ) is a G-raster, given in advance. Let G and α be as above. We say that an α-adjoint approximation scheme of the pair of groups G, G given by some rasters (K, U), (Γ Γ Γ, Ω Ω Ω) and mappings η : Obviously, if some approximation scheme is α-adjoint, ε-pairing preserving and compatible with a pair of rasters (Q, V), (∆ ∆ ∆, Υ Υ Υ), then it is ε -pairing preserving for any ε ≥ ε and compatible with any pair of rasters Q , V ≤ (Q, V), ∆ ∆ ∆ , Υ Υ Υ ≤ (∆ ∆ ∆, Υ Υ Υ).
Proof According to Corollaries 1.3.11 and 2.5.2, (in a sufficiently saturated nonstandard universe) there is a hyperfinite abelian group G and an adjoint pair of HFI approximations ζ : G → * G, ψ : G → * G. Denote by (G, G 0 , G f ), G, H 0 , H f the corresponding IMG group triplets arising from ζ , ψ , respectively. Let (Q, V), (∆ ∆ ∆, Υ Υ Υ), and α, ε satisfy the assumptions of the theorem. Then, for any G-raster (K, U) and any G-raster (Γ Γ Γ, Ω Ω Ω), which are α-adjoint, ( * K, * U) ( * Γ Γ Γ, * Ω Ω Ω), together with ζ and ψ , form an α-adjoint approximation scheme of the pair of groups * G, * G which is ε-pairing preserving with reserve. By saturation, there are an internal * G-raster K , U and an internal * G-raster Γ Γ Γ , Ω Ω Ω such that be some raster bases of the LCA groups G, G, respectively, ordered in such a way that and (ε n ) n∈N be a decreasing sequence of positive numbers ε n < 2π/3−α converging to 0. According to Theorem 2.5.5, for each i ∈ I and n ∈ N, there is an ε n -pairing preserving α-adjoint approximation scheme given by a G-raster (K in , U in ), a G-raster (Γ Γ Γ in , Ω Ω Ω in ), a finite abelian group G in and approximating maps η in : It can be easily seen that, for all j ≥ i, m ≥ n, the approximation scheme given by the rasters (K jm , U jm ), (Γ Γ Γ jm , Ω Ω Ω jm ) and the approximating maps η jm : G jm → G, φ jm : G jm → G is ε n -pairing preserving and compatible with (Q i , V i ), (∆ ∆ ∆ i , Υ Υ Υ i ), as well. Thus (Γ Γ Γ in , Ω Ω Ω in ) , (Γ Γ Γ in , Ω Ω Ω in ) , and (η in ), (φ in ) form a convergent system of α-adjoint approximation schemes for the pair of groups G, G over the directed poset (I × N, ≤) ordered by the relation (i, n) ≤ (j, m) if and only if i ≤ j and n ≤ m. The supplement on injectivity follows from the last sentence of Theorem 2.5.5.
Remarks (a) If G is metrizable and σ -compact then the same is true for G. In such a case the original raster bases can be indexed by the set I = N, as well, and the double indexing of the α-adjoint approximation schemes forming the resulting convergent system can be avoided.
(b) Similar to the proof of the implication (iv) ⇒ (ii) in Proposition 1.3.7 (see also the subsequent remark (c)), one can obtain an adjoint pair of HFI approximations η : G → * G, φ : G → * G from the convergent system of α-adjoint approximation schemes, whose existence was established in Theorem 2.5.6, by referring to the sufficient saturation of the nonstandard universe or via the ultraproduct construction applied to the approximating systems (η i : G i → G) i∈I , φ i : G i → G i∈I . The interested reader is referred to Zlatoš [49, Proposition 2.5.5, Theorem 2.5.6].
(c) The above form of Theorems 2.5.5 and 2.5.6 is not the only possible one which can be derived from Corollary 2.5.2. One could, eg, formulate the notion of compatibility with a given pair of rasters by taking care of the other two (or of all the) inclusions from Lemma 2.5.3 (b). Similarly, in Theorem 2.5.6, it is not necessary to keep the parameter α fixed; it would be possible to require that α > 0 becomes "arbitrarily small", and even guarantee that α = ε. Another example of an alternative variation can be found in Zlatoš [49, Theorems 2.5.8, 2.5.9].
The "HFI parts" of the following three examples are due to Gordon [18]; we are adding the standard counterparts mainly in order to illustrate the α-adjointness and P Zlatoš the ε-pairing preservation phenomena. In these fairly important cases, some of the previously mentioned conditions will be satisfied automatically, in an even stricter form than required by the definitions.
In the first example, building on items (a) and (b) of Example 1.3.8, we construct certain pairs of adjoint approximations of the group T and its dual group T ∼ = Z, where γ(x x x) = x x x γ for x x x ∈ T, γ ∈ Z. Similarly, the (hyper)finite cyclic group Z n is identified with its dual group Z n via the pairing γ(a) = e 2πiaγ/n , for a, γ ∈ Z n . Let us recall that Z n is still represented as the group of absolutely smallest remainders modulo n.
Example 2.5.7 If n ∈ * N ∞ then the internal homomorphism η : Z n → * T, η(a) = e 2πia/n , and the inclusion map φ : Z n → * Z give rise to the mutually dual IMG group triplets (Z n , G 0 , Z n ), Z n , {0}, Z , with normalizing multipliers d = n −1 , d = 1, respectively, where: G 0 = a ∈ Z n : a n ≈ 0 = Z For a, γ ∈ Z n we even have showing that η , φ strictly preserve the pairing on the mutually dual groups T, Z.
Passing to the standard situation, let n, k ∈ N and 0 < r ≤ α < 2π/3 be such that nr > π and 1 ≤ k < n/4. We put: In the next example we describe a pair of adjoint approximations for the self-dual group R, building on item (c) of Example 1.3.8. More precisely, R is identified with its dual group R via the pairing γ γ γ(x x x) = e ix x xγ γ γ , for x x x, γ γ γ ∈ R. The passage to differently scaled pairings γ γ γ(x x x) = e 2πix x xγ γ γ/T , with any T > 0, is straightforward.
Example 2.5.8 Let n ∈ * N ∞ , and d, d be positive infinitesimals such that both the hyperreal numbers nd , nd are infinite. Then the internal mappings η : Z n → * R, η(a) = ad , and φ : Z n → * R, φ(γ) = γd , are HFI approximations of the group R, inducing IMG group triplets (Z n , G 0 , G f ), (Z n , H 0 , H f ), where with normalizing multipliers d , d , respectively. One can easily verify that these triplets are mutually dual if and only if ndd ∈ F * R I * R. For any a, γ ∈ Z n we have (φ γ)(η a) = e iaγdd γ(a) = e 2πiaγ n and the two expressions are infinitesimally close for all a ∈ G f , γ ∈ H f if and only if: Hence this condition is equivalent to the adjointness of the HFI approximations η , φ.
Then the normalizing coefficient d = 1/nd, dual to d, is a normalizing multiplier for the triplet (Z n , H 0 , H f ), as well. Under the particular choice we even have (φ γ)(η a) = γ(a) for all a, γ ∈ Z n .
Example 2.5.9 Let G be a Hausdorff LCA group with a raster base (K i , U i ) i∈I consisting of compact open subgroups U i ⊆ K i of G. As Bohr α (X) coincides with the annihilator X ⊥ = Bohr 0 (X) for α ∈ (0, 2π/3) and any subgroup X ⊆ G, the system U ⊥ i , K ⊥ i i∈I forms a raster base of the dual group G consisting of compact open subgroups, again. Picking i ∈ I and putting U = U i , K = K i and Ω Ω Ω = K ⊥ , Γ Γ Γ = U ⊥ , it is obvious that the pairs (K, U), (Γ Γ Γ, Ω Ω Ω) are α-adjoint for each α ∈ (0, 2π/3). The quotients G = K/U and Γ Γ Γ/Ω Ω Ω are finite abelian groups and the latter can be canonically identified with the dual group G. Any right inverse map η : G → K ⊆ G to the canonical projection ζ : K → K/U is a (K, U) approximation of G, and similarly, any right inverse map φ : G → Γ Γ Γ ⊆ G to the canonical projection ξ : Γ Γ Γ → Γ Γ Γ/Ω Ω Ω is a (Γ Γ Γ, Ω Ω Ω) approximation of G. Then one can easily verify by a straightforward computation that, for any a ∈ G, γ ∈ G: show that the approximation scheme given by the α-adjoint rasters (K, U), (Γ Γ Γ, Ω Ω Ω) and the mappings η , φ is compatible with these very rasters, hence it is a pairing preserving α-adjoint approximation scheme for the pair G, G.

P Zlatoš
In Gordon [18], pairs of adjoint HFI approximations for τ -adic solenoids Σ τ and their dual groups of τ -adic rationals are also described. These, as well as the corresponding finite approximations, can be constructed combining some ideas from Examples 2.5.7-2.5.9.
In view of the supplements on injectivity after Corollary 2.5.2 and in Theorems 2.5.5 and 2.5.6, as well as of the last three examples, it is natural to formulate the following conjecture. We prefer to state it in the more concise nonstandard form. The standard translation can be readily obtained from Theorem 2.5.5.
Conjecture Let G be a Hausdorff LCA group. Then there exists a hyperfinite abelian group G and injective mappings η : G → * G, φ : G → * G forming an adjoint pair of HFI approximations of the mutually dual groups G, G, respectively.
Let us conclude by a brief discussion of the roles of standard and nonstandard methods in the proof of the Adjoint Approximation Scheme Theorem 2.5.5. Our starting point was the proof of the existence of arbitrarily good (standard) finite approximations for any single LCA group G (Theorem 1.3.10) from which we derived the existence of (nonstandard) hyperfinite approximations of G by means of the Transfer Principle (Corollary 1.3.11). Next we proved another nonstandard result, namely the existence of an adjoint approximation to any HFI approximation of G (Theorem 2.5.1 and Corollary 2.5.2). Both in their formulation and proof the (inherently nonstandard) Gordon's Conjecture 1 (Theorem 2.2.4) and the Triplet Duality Theorem 2.2.5 were crucial. Finally we turned back, deriving the existence of an arbitrarily precise pairing preserving adjoint pair of (standard) finite approximations of the mutually dual groups G, G (Theorem 2.5.5) from the nonstandard Corollary 2.5.2, using the Transfer Principle in the "opposite direction".