Sets of transfer times with small densities

We consider in this paper the set of transfer times between two measurable subsets of positive measures in an ergodic probability measure-preserving system of a countable abelian group. If the lower asymptotic density of the transfer times is small, then we prove this set must be either periodic or Sturmian. Our results can be viewed as ergodic-theoretical extensions of some classical sumset theorems in compact abelian groups due to Kneser. Our proofs are based on a correspondence principle for action sets which was developed previously by the first two authors.


Introduction
Throughout this paper, we shall assume that -G is a countable and discrete abelian group.
-(X, µ) is a standard probability measure space, endowed with an ergodic probability measure-preserving action of G. In other words, (X, µ) is an ergodic Borel G-space.
-(F n ) is a sequence of finite subsets of G with the property that for every bounded measurable function ϕ on X, there exists a µ-conull subset X ϕ ⊂ X such that The set of transfer times R A,B is defined by We set R A = R A,A , and refer to R A as the set of return times to the set A. If we wish to emphasize the dependence on the measure µ, we write R µ A,B and R µ A respectively.
The aim of this paper is to establish various lower bounds on d(R A,B ), and to discuss when these lower bounds are attained. As we shall see in the proofs below, these questions are closely related to direct and inverse theorems for product sets with small doubling, which is an active line of research in additive combinatorics.
Before we state our main results, we make a few preliminary remarks. Firstly, our assumption (1.1) on the sequence (F n ) readily implies that the lower asymptotic density of R A,B is strictly positive for all measurable subsets A and B of X with positive measures. Secondly, (i) If µ(A) + µ(B) > 1, then µ(A ∩ g −1 B) > 0 for all g, whence R A,B = G.
(ii) If µ(A) + µ(B) = 1, then either R A,B = G, or µ(A ∩ g −1 o B) = 0 for some g o . In the latter case, B = g −1 o A c modulo µ-null sets, so if denote by H the µ-essential stabilizer of A, then R A,B = G g −1 o H. In order to arrive at non-trivial results about sets of transfer times, we shall for the rest of the paper, always assume that the measurable subsets A and B in X satisfy (1.4) µ(A) + µ(B) < 1.

1.1.
Main results. -Our first theorem roughly asserts that if the lower asymptotic density of R A is small enough, then the set of transfer times R A is in fact a subgroup of G. (ii) If d(R A ) < 3 Our second theorem in particular tells us that a measurable subset A ⊂ X with positive µ-measure for which (ii) in Theorem 1.3 holds must be rather special. To be able to state the full result, we need the following notation of control.
Definition 1.4. -Let (Y, ν) be an ergodic Borel G-space and let π : (X, µ) → (Y, ν) be a G-factor map. If (A, B) and (C, D) are pairs of measurable subsets with positive measures in X and Y respectively, then we say that (C, If we wish to emphasize the dependence on π, we say that (C, D) π-controls (A, B).
Remark 1.6. -Theorem 1.5 can be viewed as an ergodic-theoretical extension of Kneser's celebrated inverse theorem [4] for the lower asymptotic density of sumsets in (N, +), which roughly asserts that if d(A + B) < d(A) + d(B) for two subsets A and B of N, where d denotes the lower asymptotic density with respect to the sequence ([1, n]), then A + B is periodic (modulo a finite set). The connection between sumsets and sets of transfer times will be discussed in more details below. Theorem 1.5 also tells us that an ergodic Borel G-space which admits a pair (A, B) of measurable subsets with positive measures with a small set of transfer times (in the sense that the inequality d(R A,B ) < µ(A) + µ(B) holds) must have a non-trivial periodic G-factor. We recall that a Borel G-space is called totally ergodic if every finiteindex subgroup G o < G acts ergodically. Such Borel G-spaces cannot have non-trivial periodic G-factors, and thus we conclude the following corollary from Theorem 1.5. Our third theorem characterizes exactly when the lower bound in Corollary 1.7 is attained (assuming that the action is totally ergodic).
(ii) a G-factor σ : (X, µ) → (T, m T ), where m T denotes the normalized Haar measure on T and G acts on T via η.
, modulo at most two cosets of the subgroup ker η. Remark 1.9. -Conversely, it is not hard to show that if a pair (A, B) of measurable subsets in X is σ-controlled by a pair (I o , J o ) as in the theorem above, then and thus Theorem 1.8 really provides a complete characterization of when the lower bound in Corollary 1.7 is attained (assuming total ergodicity).
It is not hard to check that where A = {C ∈ 2 Z | 0 ∈ C}, and such that the set of return times R A projects onto every finite quotient of Z. In particular, R A cannot be a subgroup of Z, nor can it be contained in a subgroup of Z. Here, the exact choice of the sequence (F n ) in Z is not so important; for simplicity, we can assume that F n = [1, n] for all n 1.
The construction of µ goes along the following lines. Given positive real numbers δ and ε, we choose 0 < η < 1 such that 1 + ε = (1 + η)/(1 − η), and we pick a strictly increasing sequence (p k ) of prime numbers such that For every k 1, we denote by µ k the uniform probability measure on the Z-orbit of the subgroup p k Z in 2 Z and we note that µ k (A) = 1/p k . We now define which is clearly a Z-invariant non-ergodic Borel probability measure on 2 Z . One readily checks that and, thus, by (1.5) and the choice of η, Clearly, R A projects onto every finite quotient of Z, which finishes our construction. . We stress however that we do not need to assume that the sequence (F n ) is Følner (asymptotically invariant) in G. For instance, in the case of (Z, +), our results above also apply to the sparse sequence F n = k √ 2 + k 5/2 | k = 1, . . . , n , n 1, which is far from being a Følner sequence in Z, but nevertheless satisfies (1.1) by [2].
Example 1.4 ("Non-conventional" lower asymptotic density). -If Z (X, µ) is totally ergodic, then the sequence (F n ) of squares F n = {k 2 | k = 1, . . . , n}, for n 1, satisfies (1.1) (the almost sure convergence follows from the work of Bourgain [3], while the identification of the limit -for totally ergodic actions -follows from the equidistribution (modulo 1) of the sequence (n 2 α), for irrational α). In particular, by Corollary 1.7 we have for all measurable subsets A, B ⊂ X with positive µ-measures.

On ergodic actions with small doubling
Remark 1.11. -We note that if the action is C-doubling, then it must also be C -doubling for every C C.
It seems natural to ask about the structure of C-doubling actions. The following theorem provides a complete characterization of such actions.
-if the identity component K o of K has finite index, then the action is 2-doubling.
Remark 1.13. -Theorem 1.12 in particular asserts that an ergodic action is C-doubling for some C 1 if and only if it has an infinite Kronecker factor (see e.g. [1] for definitions).
The same line of argument as the one leading up to Theorem 1.12 also proves the following result, whose proof we leave to the reader. We recall that G (X, µ) is weakly mixing if the diagonal action G (X × X, µ ⊗ µ) is ergodic.
is not weakly mixing.

1.4.
A brief outline of the proofs. -Our first observation is that for any two measurable subsets A and B of X with positive µ-measures, there is a measurable µ-conull subset X 1 of X such that where A x and B x are the return times of the point x to the sets A and B (see Section 2.1 below for notation). We then observe in Lemma 2.2 that for some measurable µ-conull subset X 2 ⊂ X, for all x ∈ X 2 , which puts us in the framework of our earlier paper [1]. We combine some of the key points of this paper in Lemma 2.4 below, the outcome of which is that there exist -a compact and metrizable abelian group K with Haar probability measure m K and a homomorphism τ : In particular, and if A = B, then we can take I = J. In the settings of Theorem 1.3, Theorem 1.5 and Theorem 1.8, we see that respectively. At this point, we use some classical results [5] of Kneser for sumsets in compact abelian groups, to conclude that the pair (I, J) is "reduced" to a nicer pair (I o , J o ) in a much "smaller" quotient group Q of K (see Definition 2.6 for details).
The point of all this is that the transfer times R A,B is contained in the transfer times between I o and J o , which is equal to the set η −1 (J o I −1 o ). Here η : G → Q is the composition of τ with the quotient map from K to Q. To prove that the sets actually coincide, we shall use the overshoot relation This inequality is proved in Proposition 2.7. It turns out that in the settings of the theorems above, the sets I o and J o have the property that the m Q -measure of the intersection 1.5. Ergodic actions of semi-groups. -Our definition of transfer times between two sets makes sense also for actions by non-invertible maps. Suppose that S is a countable abelian semigroup, sitting inside a countable abelian group G. If S acts ergodically by measure-preserving maps on a standard probability measure space (X, µ), then, under some technical assumptions (see e.g. [6] for more details in the general setting), one can construct a so called natural extension ( X, µ) of the S-action, which is a measure-preserving G-action, together with a measurable S-equivariant map ρ : X → X, mapping µ to µ. It is not hard to see that if we set where the transfer times R A, B are measured with respect to µ. We can now apply our results above to the G-action on the natural extension ( X, µ) (which is ergodic if and only if the semi-group action S (X, µ) is), and conclude the same results for the S-action. We leave the details to the interested reader.
Acknowledgements. -The authors thank the anonymous referees for their comments. I.S. is grateful to SMRI and the School of Mathematics and Statistics at Sydney University for funding his visit and for their hospitality. M.B. and A.F. wish to thank the organizers of the MFO workshop "Groups, dynamics and approximation", during which parts of this paper were written, for the invitation.

Preliminaries
2.1. Transfer times and action sets. -Given a subset D of X and x ∈ X, we define the set of return time of x to D by In particular, 2.2. Transfer times as difference sets. -Let D be a measurable subset of X, and define We note that D e = g∈G gD c and D ne = GD. In particular, D e and D ne are both measurable and G-invariant. Since µ is assumed to be ergodic, we conclude that  Then X 1 is a G-invariant measurable µ-conull subset of X and Proof. -Measurability and G-invariance of X 1 is clear, and µ-conullity of X 1 readily follows from applying (2.2) and (2.3) to the sets Indeed, µ(D(g)) > 0 if and only if g ∈ R A,B , and by (2.1), we have Note that for every x ∈ X, which finishes the proof.

Generic points.
-We recall our assumptions on the sequence (F n ) of finite subsets of G: For every bounded measurable function ϕ on X, there exists a µ-conull subset X ϕ ⊂ X such that The points in X ϕ are said to be generic with respect to µ, ϕ and the sequence (F n ).
Lemma 2.2. -Let A and B be two measurable subsets of X with positive µ-measures. Then there exists a measurable µ-conull subset X 2 ⊆ X such that for all x ∈ X 2 .
Proof. -Given a subset L ⊂ G, we define We note X L is a measurable µ-conull subset of X and for every x ∈ X L , We now set X 2 = L X L , where the intersection is taken over the countable set of all finite subsets of G. Then X 2 is a measurable µ-conull subset of X, and for every x ∈ X 2 and for every finite subset L of A x , we have Since µ is σ-additive and L ⊂ A x is an arbitrary finite set, we can now conclude that x for all x ∈ X 1 , and thus, since G is abelian,  Proof. -By Lemma 2.2, there is a measurable µ-conull subset X 2 of X such that for all x ∈ X 2 . Since the roles of A and B are completely symmetric, this proves the corollary.

A correspondence principle for action sets. -
The key ingredient in the proofs of Theorem 1.3, Theorem 1.5 and Theorem 1.8 is the following merger of a series of observations made by the first two authors in [1]. We outline the anatomy of this merger in the proof below. The rough idea is that action sets in an arbitrary ergodic G-action can be controlled by sets in an isometric factor (that is to say, a compact group, on which G acts by translations via a homomorphism from G into the compact group with dense image).
-a compact and metrizable abelian group K with Haar probability measure m K , a homomorphism τ : G → K with dense image, and two measurable subsets I and J of K, In the case when A = B, we can take I = J. Finally, if G (X, µ) is totally ergodic, then K must be connected.
Remark 2.5. -If I and J are Borel measurable subsets of K, then their difference set I −1 J might fail to be Borel measurable. However, since I −1 J is the image of the Borel measurable subset I × J in K × K under the continuous map (k 1 , k 2 ) → k −1 1 k 2 , we see that I −1 J is an analytic set, so in particular measurable with respect to the completion of the Borel σ-algebra of K with respect to m K , and thus the expression A × B)), for all x ∈ X 3 . By [1, Th. 5.1], there exist -a measurable G-invariant µ-conull subset X 3 ⊂ X, -a compact and metrizable abelian group K with Haar probability measure m K and a homomorphism τ : G → K with dense image, -a G-equivariant measurable map π : X 3 → K such that π * (µ| X 3 ) = m K , where G acts on K via τ , -two measurable subsets I and J of K, It follows from the proof of [1, Th. 5.1] that if A = B, then we can take I = J. Since the set X 3 is G-invariant, we see that . Let X 3 := X 3 ∩ X 3 and note that X 3 is G-invariant and µ-conull. Thus the proof is finished modulo our assertion about total ergodicity. Suppose that K is not connected.
Then there is an open subgroup U of K, and G o = τ −1 (U ) is a finite-index subgroup of G. We note that C := π −1 (U ) is a G o -invariant measurable subset of X, with positive µ-measure, but which cannot be µ-conull, since it does not map onto K under π (modulo µ-null sets). We conclude that G (X, µ) is not totally ergodic.

2.5.
Putting it all together. -Let K and Q be compact groups with Haar probability measures m K and m Q respectively and suppose that there is a continuous homomorphism p from K onto Q. This notion is quite useful when we now summarize our discussion above.
In the case when A = B, we can take I = J.
Proof. -By Lemma 2.4, we can find a G-invariant measurable µ-conull subset X 3 ⊆ X, a compact and metrizable abelian group K with Haar probability measure m K , a homomorphism τ : G → K with dense image, and two measurable subsets I and J of K, a G-equivariant measurable map π : X 3 → K such that π * (µ| X3 ) = m K , where G acts on K via τ , such that (2.6) A ∩ X 3 ⊆ π −1 (I) and B ∩ X 3 ⊆ π −1 (J) and for all x ∈ X 3 . Furthermore, by Lemma 2.1 and Lemma 2.2, there exist measurable µ-conull subsets X 1 and X 2 of X such that ) and, for every g / ∈ R A,B , for all x ∈ X 1 ∩ X 2 . In particular, since X 1 ∩ X 2 ∩ X 3 is a µ-conull subset of X, and thus non-empty, we have Let us now assume that Q is a compact group, p : K → Q is a continuous surjective homomorphism and I o and J o are measurable subsets of Q such that (I, J) reduces to (I o , J o ). We recall that this means that , for all x ∈ X 3 . We note that we can write , for all x ∈ X 3 , where σ = p • π, and thus (2.8) for all x ∈ X 3 . The map σ is a G-factor map from (X, µ) to (Q, m Q ), where G acts on Q via τ p = p•τ , and it follows from (2.6) that It remains to prove that To prove the inclusion, we first note that since X 3 is G-invariant, we have , for all x ∈ X 3 . To prove (2.10), we recall from (2.7) that if g / ∈ R A,B , then whence, by (2.9), which finishes the proof.
2.6. Classical product set theorems in compact groups. -We shall use the following two results about product sets in compact groups due to Kneser in his very influential paper [5]. Then JI −1 is a clopen subset of K, and there exist -a finite group Q and a homomorphism p from K onto Q. Corollary 2.9. -Let K be a compact and metrizable abelian group with Haar probability measure m K and assume that I is a measurable subset of K with positive m K -measure such that Then there exist a finite group Q, a surjective homomorphism p : K → Q and a point q ∈ Q such that (I, I) reduces to ({q}, {q}) with respect to p. In particular, II −1 is an open subgroup of K.
Proof. -By Theorem 2.8, there exist a finite group Q, a homomorphism p from K onto Q and a subset I o of Q, such that If K is connected and non-trivial, then there are no proper clopen subsets of K, whence the assumed upper bound in Theorem 2.8 can never occur. In the connected case, Kneser further characterized the pairs of measurable subsets of the group for which the lower bound in Corollary (2.10) is attained. We denote by T the group R/Z endowed with the quotient topology. In the case (3.1), which corresponds to Theorem 1.3, we can take I = J, and thus and in the case (3.2), which corresponds to Theorem 1.5, we have In both cases, Theorem 2.8 tells us that there exist a finite group Q, a continuous surjective homomorphism p : K → Q and a pair (I o , J o ) of subsets of Q such that (I, J) reduces to (I o , J o ) with respect to p. By Proposition 2.7, this implies that In the case (3.1), Corollary 2.9 further asserts that I o = J o = {q} for some point q ∈ Q, whence I o I −1 o = e Q and thus we can conclude from above that R A ⊂ G o := ker τ p , and Since Q is finite, G o has finite index in G and for every g ∈ G o R A , we have The last inequality clearly contradicts (3.3), so we conclude that G o = R A , which finishes the proof of Theorem 1.3.
In the case of (3.2), Theorem 2.8 asserts that the pair By Proposition 2.7, we have which clearly contradicts (3.4). We conclude that τ −1  Furthermore, since G (X, µ) is totally ergodic, K must be connected. In particular, by Corollary 2.10,

Proof of Theorem 1.12
Let us first assume that G (X, µ) is C-doubling for some C 1. Then, for every n 1, there is a measurable subset A n ⊂ X such that 0 < µ(A n ) < 1 n and d(R An ) Cµ(A n ) < C n .
To avoid trivialities, we shall from now on assume that n > C. By Lemma 2.4, we can find a (non-trivial) compact metrizable group K n , a homomorphism η n : G → K n with dense image, a G-factor map π n : (X, µ) → (K n , m Kn ) and a measurable subset I n ⊂ K n such that A n ⊂ π −1 n (I n ) modulo null sets and m Kn (I −1 n I n ) C n , for all n 1.
In particular, m Kn (I n ) m Kn (I −1 n I n ) C n . Let K denote the closure in n K n of the diagonally embedded subgroup {(η n (g)) | g ∈ G}. We note that π = (π n ) : (X, µ) → (K, m K ) is a G-factor map, where G acts on K via η = (η n ). Since the pull-backs to K of the sets I n provide measurable subsets of K with arbitrarily small m K -measures, we see that K must be infinite.
Let us now assume that there exist (i) an infinite compact metrizable group K and a homomorphism η : G → K with dense image.
(ii) a G-factor σ : (X, µ) → (K, m K ), where m K denotes the normalized Haar measure on K and G acts on K via η.
We wish to prove that (X, µ) is C-doubling for some C 1. Since G (K, m K ) is a G-factor of (X, µ), it is clearly enough to prove that G (K, m K ) is C-doubling. If K o has infinite index in K, then K/K o is an infinite totally disconnected group, and thus we can find a decreasing sequence (U n ) of open subgroups of K with m K (U n ) = 1/[K : U n ] < 1/n for all n. Since R Un = η −1 (U n ), we have d(R Un ) = 1 [K : U n ] = m K (U n ), for all n 1, which shows that G (X, µ) is 1-doubling (we are using here that the sequence (F n ) also satisfy (1.1) for all bounded measurable functions on K). If K o has finite index in K, then K o is an open subgroup, and thus has positive m K -measure. Fix a nontrivial continuous character χ : K o → T, and note that by connectedness, χ is onto. Set Then, m K (I n ) = m K (K o )/n for all n, and it is not hard to show that d(R In ) = d(η −1 (I n − I n )) 2m K (I n ), for all n, whence G (X, µ) is 2-doubling.