Preliminaries

2.2. Transfer times as difference sets. — LetDbe a measurable subset ofX, and define measurable andG-invariant. Sinceµis assumed to be ergodic, we conclude that

(2.2) µ(D^e) = 1 ifµ(D) = 0

and

(2.3) µ(D^ne) = 1 ifµ(D)>0.

Lemma2.1. — LetAandB be two measurable subsets ofX with positiveµ-measures, and define

ThenX1 is aG-invariant measurable µ-conull subset ofX and RA,B=BxA⁻¹_x , for all x∈X1.

Proof. — Measurability andG-invariance ofX₁is clear, andµ-conullity ofX₁readily follows from applying (2.2) and (2.3) to the sets

D(g) :=A∩g⁻¹B, forg∈G.

2.3. Generic points. — We recall our assumptions on the sequence (F_n) of finite subsets ofG: For every bounded measurable functionϕonX, there exists aµ-conull subsetXϕ⊂X such that

Lemma2.2. — LetAandB be two measurable subsets ofX with positiveµ-measures.

Then there exists a measurableµ-conull subsetX2⊆X such that µ(A⁻¹_x B)6d(RA,B), for all x∈X2. Furthermore, for every finite subsetL ofG,

d(L⁻¹Ax) =µ(L⁻¹A) and d(L⁻¹Bx) =µ(L⁻¹B),

and

LX_L, where the intersection is taken over the countable set of allfinite subsets of G. ThenX₂⁰ is a measurableµ-conull subset of X, and for every x∈X₂⁰ and for every finite subsetL ofAx, we have

d(A⁻¹_x B_x)>d(L⁻¹B_x) =µ(L⁻¹B).

Sinceµisσ-additive andL⊂A_xis an arbitrary finite set, we can now conclude that d(A⁻¹_x Bx)>µ(A⁻¹_x B) for allx∈X₂⁰.

By Lemma 2.1, there exists a measurableµ-conull subsetX1⊆X such thatRA,B= B_xA⁻¹_x for allx∈X₁, and thus, sinceGis abelian, limits of each term exist (the last identity follows from the fact thatµ(A∩g⁻¹B) = 0 ifg /∈RA,B). Sincex∈X2 is arbitrary, this finishes the proof.

Corollary2.3. — For all measurable subsetsA andB of X, we have d(RA,B)>max(µ(A), µ(B)).

Proof. — By Lemma 2.2, there is a measurableµ-conull subsetX2ofX such that d(RA,B)>µ(A⁻¹_x B)>µ(B), for allx∈X2.

Since the roles ofAandB are completely symmetric, this proves the corollary.

2.4. 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).

Lemma2.4. — Let A and B be measurable subsets of X with positive µ-measures.

Then there exist

– aG-invariant measurableµ-conull subsetX3⊆X,

– a compact and metrizable abelian group K with Haar probability measure mK, a homomorphismτ :G→K with dense image, and two measurable subsets I andJ of K,

– aG-equivariant measurable mapπ:X₃→Ksuch thatπ_∗(µ|_X3) =m_K, whereG acts onK viaτ,

such that

A∩X3⊆π⁻¹(I) and B∩X3⊆π⁻¹(J) and

µ(A⁻¹_x B) =m_K(I⁻¹J) and A⁻¹_x (B∩X₃)⊆π⁻¹(π(x)I⁻¹J),

for allx∈X₃. In the case when A=B, we can takeI =J. Finally, if Gy(X, µ) is totally ergodic, thenK must be connected.

Remark2.5. — IfI andJ are Borel measurable subsets ofK, then their difference setI⁻¹J might fail to be Borel measurable. However, sinceI⁻¹J is the image of the Borel measurable subsetI×J inK×Kunder the continuous map(k1, k2)7→k⁻¹₁ k2, we see thatI⁻¹J is an analytic set, so in particular measurable with respect to the completion of the Borelσ-algebra of K with respect tomK, and thus the expression m_K(I⁻¹J)is well-defined.

Proof. — By [1, Lem. 5.3], there exists a G-invariant measurable µ-conull subset X₃⁰ ⊂X such that

µ(A⁻¹_x B) =µ⊗µ(G(A×B)), for allx∈X₃⁰. By [1, Th. 5.1], there exist

– a measurableG-invariantµ-conull subsetX₃⁰⁰⊂X,

– a compact and metrizable abelian groupK with Haar probability measuremK

and a homomorphismτ :G→K with dense image,

– aG-equivariant measurable mapπ:X₃⁰⁰→Ksuch thatπ_∗(µ|_X⁰⁰

3) =m_K, whereG acts onKviaτ,

– two measurable subsetsI andJ ofK, such that

µ⊗µ(G(A×B)) =m_K(I⁻¹J) and

A∩X₃⁰⁰⊂π⁻¹(I) and B∩X₃⁰⁰⊂π⁻¹(J).

It follows from the proof of [1, Th. 5.1] that ifA=B, then we can takeI=J. Since the setX₃⁰⁰isG-invariant, we see that

Ax⊂π⁻¹(I)x=τ⁻¹(Iπ(x)⁻¹), for allx∈X₃⁰⁰, whence

A⁻¹_x (B∩X₃⁰⁰)⊂A⁻¹_x π⁻¹(J) =π⁻¹(τ(Ax)⁻¹J)⊂π⁻¹(π(x)I⁻¹J).

LetX3 :=X₃⁰ ∩X₃⁰⁰and note that X3 isG-invariant andµ-conull. Thus the proof is finished modulo our assertion about total ergodicity. Suppose thatKis not connected.

Then there is an open subgroupU ofK, andGo=τ⁻¹(U)is a finite-index subgroup of G. We note that C := π⁻¹(U) is a Go-invariant measurable subset of X, with positive µ-measure, but which cannot be µ-conull, since it does not map onto K underπ(moduloµ-null sets). We conclude thatGy(X, µ)is not totally ergodic.

2.5. Putting it all together. — LetK andQbe compact groups with Haar prob-ability measures mK and mQ respectively and suppose that there is a continuous homomorphismpfromK ontoQ.

Definition2.6(Pair reduction). — Let(I, J)and(I_o, J_o)be two pairs of measurable subsets of K and Q respectively. We say that (I, J) reduces to (Io, Jo) with respect topif

I⊂p⁻¹(I_o) and J ⊂p⁻¹(J_o) and m_K(J I⁻¹) =m_Q(J_oI_o⁻¹).

This notion is quite useful when we now summarize our discussion above.

Proposition2.7(A correspondence principle for transfer times). — Let A andB be measurable subsets ofX with positiveµ-measures. Then there exist

– a compact and metrizable abelian group K with Haar probability measurem_K, – a homomorphism τ:G→K with dense image,

– a pair (I, J)of measurable subsets ofK, which satisfy

µ(A)6mK(I) and µ(B)6mK(J) and mK(J I⁻¹)6d(RA,B).

Furthermore, suppose that Q is a compact group and p : K → Q is a continuous surjective homomorphism. If (Io, Jo) is a pair of measurable subsets of Q such that (I, J)reduces to (Io, Jo) with respect top, then

RA,B⊆τ_p⁻¹(J_oI_o⁻¹),

whereτp=p◦τ, and for allg∈τ_p⁻¹(JoI_o⁻¹)rRA,B, we have

µ(A) +µ(B)6mQ(Io) +mQ(Jo)−mQ(Io∩τp(g)⁻¹Jo),

Moreover, there exists a G-factor map σ : (X, µ) → (Q, m_Q), where G acts on Q via τp, such that

A⊆σ⁻¹(Io) and B⊆σ⁻¹(Jo), moduloµ-null sets.

In the case when A=B, we can takeI=J.

Proof. — By Lemma 2.4, we can find a G-invariant measurable µ-conull subset X3⊆X, a compact and metrizable abelian group K with Haar probability mea-sure mK, a homomorphism τ : G → K with dense image, and two measurable subsets I and J of K, a G-equivariant measurable map π : X₃ → K such that π_∗(µ|X₃) =mK, where Gacts onK viaτ, such that

(2.6) A∩X3⊆π⁻¹(I) and B∩X3⊆π⁻¹(J)

and

m_K(J I⁻¹)6µ(A⁻¹_x B) and A⁻¹_x (B∩X₃)⊆π⁻¹(π(x)I⁻¹J),

for all x∈X3. Furthermore, by Lemma 2.1 and Lemma 2.2, there exist measurable µ-conull subsetsX1andX2 ofX such that

RA,B=B_xA⁻¹_x and µ(A⁻¹_x B)6d(RA,B) and, for everyg /∈RA,B,

(2.7) d(Ax∪g⁻¹Bx) =µ(A) +µ(B) =µ(A∪g⁻¹B),

for allx∈X₁∩X₂. In particular, sinceX₁∩X₂∩X₃is a µ-conull subset ofX, and thus non-empty, we have

µ(A)6mK(I) and µ(B)6mK(J) and mK(J I⁻¹)6d(RA,B).

Let us now assume thatQis a compact group,p:K→Qis a continuous surjective homomorphism and Io andJo are measurable subsets ofQsuch that(I, J) reduces to(Io, Jo). We recall that this means that

I⊂p⁻¹(I_o) and J ⊂p⁻¹(J_o) and m_K(J I⁻¹) =m_Q(J_oI_o⁻¹).

Hence,J I⁻¹⊂p⁻¹(JoI_o⁻¹), and

mQ(JoI_o⁻¹)6µ(A⁻¹_x B) and A⁻¹_x (B∩X3)⊆π⁻¹(π(x)p⁻¹(JoI_o⁻¹)), for allx∈X₃. We note that we can write

π⁻¹(π(x)p⁻¹(JoI_o⁻¹)) =σ⁻¹(σ(x)JoI_o⁻¹), for allx∈X₃, whereσ=p◦π, and thus

(2.8) A⁻¹_x (B∩X3)⊆σ⁻¹(σ(x)JoI_o⁻¹), for allx∈X3.

The mapσis aG-factor map from(X, µ)to(Q, mQ), whereGacts onQviaτp=p◦τ, and it follows from (2.6) that

(2.9) A∩X₃⊂σ⁻¹(I_o) and B∩X₃⊂σ⁻¹(J_o).

It remains to prove that

RA,B⊆τ_p⁻¹(J_oI_o⁻¹), and that for everyg∈τ_p⁻¹(JoI_o⁻¹)rRA,B, we have

(2.10) µ(A) +µ(B)6mQ(Io) +mQ(Jo)−mQ(Io∩τp(g)⁻¹Jo).

To prove the inclusion, we first note that sinceX₃ isG-invariant, we have (A⁻¹_x (B∩X3))x=A⁻¹_x Bx⊂σ⁻¹(σ(x)JoI_o⁻¹)x

=τ_p⁻¹(σ(x)J_oI_o⁻¹σ(x)⁻¹) =τ_p⁻¹(J_oI_o⁻¹), for allx∈X3. To prove (2.10), we recall from (2.7) that ifg /∈RA,B, then

d(Ax∪g⁻¹Bx) =µ(A) +µ(B) =µ(A∪g⁻¹B),

whence, by (2.9),

d(A_x∪g⁻¹B_x) =µ(A) +µ(B) =µ(A∪g⁻¹B)

6µ(σ⁻¹(I_o)∪g⁻¹σ⁻¹(J_o)) =m_Q(I_o∪τ_p(g)⁻¹J_o)

=mQ(Io) +mQ(Jo)−mQ(Io∩τp(g)⁻¹Jo),

which finishes the proof.

2.6. Classical product set theorems in compact groups. — We shall use the fol-lowing two results about product sets in compact groups due to Kneser in his very influential paper [5].

Theorem2.8([5, Satz 1]). — Let K be a compact and metrizable abelian group with Haar probability measuremK and suppose thatI andJ are measurable subsets of K with positivemK-measures such that

mK(J I⁻¹)< mK(I) +mK(J).

ThenJ I⁻¹ is a clopen subset of K, and there exist

– a finite group Qand a homomorphism pfrom K ontoQ.

– a pair (I_o, J_o)of subsets of Qwith

mQ(JoI_o⁻¹) =mQ(Jo) +mQ(Io)−mQ({eQ}),

such that (I, J)reduces to(Io, Jo)with respect top. IfI=J, we can takeIo=Jo.

Corollary2.9. — LetK be a compact and metrizable abelian group with Haar prob-ability measure mK and assume that I is a measurable subset of K with positive m_K-measure such that

mK(II⁻¹)<3

2mK(I).

Then there exist a finite groupQ, a surjective homomorphismp:K→Qand a point q ∈Q such that (I, I) reduces to ({q},{q})with respect to p. In particular, II⁻¹ is an open subgroup ofK.

Proof. — By Theorem 2.8, there exist a finite groupQ, a homomorphismpfrom K ontoQand a subset Io ofQ, such that

I⊂p⁻¹(Io) and II⁻¹=p⁻¹(IoI_o⁻¹) and mQ(IoI_o⁻¹) = 2mQ(Io)−mQ({eQ}).

Sincem_K(II⁻¹)< ³₂m_K(I), we conclude that

(2.11) m_Q(I_oI_o⁻¹) = 2m_Q(I_o)−m_Q({e_Q})< 3

2m_Q(I_o),

whence mQ(IoI_o⁻¹) < ³₂mQ({eQ}). Since Io is non-empty, we conclude that IoI_o⁻¹

must be a point. HenceIo={q} for someq∈Q.

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.

Corollary2.10. — LetKbe a compact, metrizable and connected abelian group with Haar probability measuremK. Then, for all measurable subsetsI andJ of K,

mK(J I⁻¹)>min 1, mK(I) +mK(J) .

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 byTthe groupR/Zendowed with the quotient topology.

Theorem2.11([5, Satz 2]). — Let K be a compact, metrizable and connected abelian group with Haar probability measurem_K. IfI andJ are measurable subsets ofKsuch that

m_K(J I⁻¹) =m_K(I) +m_K(J)61, then there exist

– a continuous homomorphismpfromK ontoT, – closed intervalsIo andJo in Twith

m_T(Io) =mK(I) and m_T(Jo) =mK(J), such that (I, J)reduces to(Io, Jo)with respect top.