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Product sets in sets of returns and positivity of symmetric ergodic averages

T0 review · 4 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The paper proves that sets of measurable returns of a positive-measure set in a $G\times G$ action always contain a Cartesian square $B\times B$, and that in finitely generated nilpotent groups the analogous one-variable product $B\cdot…

desk verdict A substantial and largely sound generalization of Bergelson's difference-set theorems to arbitrary countable amenable groups and several non-abelian classes, with the only load-bearing concern being the polynomial section's reliance on a modified reading of Zorin-Kranich's IP-polynomial theorem. read the letter →

arxiv 2608.02873 v1 pith:AQ3CSZC7 submitted 2026-08-03 math.DS math.COmath.GR

Vitaly Bergelson , Saúl Rodríguez-Martín This is my paper · ORCID
classification math.DSmath.COmath.GR MSC 37A1537A3005D1011B05
keywords setsofreturnsproductupperBanachdensityamenablegroupsnilpotentsymmetricergodicaveragespolynomialrecurrenceFølnersequences
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper establishes that, for measure-preserving actions of a countable group $G$, the set of group elements that return a positive-measure set to itself is highly structured: in a $G\times G$ action it always contains a Cartesian product $B\times B$ where $B$ is large. When $G$ is amenable, this becomes a purely combinatorial statement: every set $A\subseteq G\times G$ with positive upper Banach density has $B\times B\subseteq AA^{-1}$ for some $B$ whose density along any prescribed F\u00f8lner sequence is at least the square of the density of $A$. The paper also proves the harder one-variable analogue, $BB\subseteq AA^{-1}$, for broad classes of amenable groups, including finitely generated nilpotent groups, semidirect products of abelian groups, and matrix groups over fields that are unions of finite fields; for nilpotent groups the bound is explicit: the relevant symmetric ergodic average is at least $2^{-n(n-1)/2}\mu(Y)^2$.

What carries the argument

The load-bearing mechanism is a general 'product sets in sets of measurable returns' theorem (Theorem 3.1): if functions $\phi_i,\psi_i:G\to H$ are given and the averaged measure of $\bigcap_i T_{\phi_i(g)}Y\cap T_{\psi_i(g)}^{-1}Y$ has limit $\lambda$, then there is $B\subseteq G$ with $d_F(B)\ge\lambda$ and $\bigcup_{i,j}\psi_i(B)\phi_j(B)$ contained in the return set of $Y$. Its proof rests on an intersectivity lemma (Lemma 2.2), which extracts a set $B$ of density at least $\lambda$ from a family of measurable sets with average intersection measure $\lambda$. Combined with the von Neumann mean ergodic theorem for amenable groups (which gives limit at least $\mu(Y)^2$) and, for nilpotent groups, with Zorin-Kranich's IP-polynomial multiple recurrence theorem, this yields both the Cartesian-square and product-set conclusions. The nilpotent bounds further use a Malcev embedding into $UT_n(\mathbb{Z})$ and a carefully chosen subgroup whose squaring map is almost a bijection onto a F\u00f8lner sequence.

What would settle it

Exhibit a single step in Zorin-Kranich's proof (for instance Proposition 2.22, Lemma 4.20, or the limit interchange in Section 5) where the condition $\alpha\cap\beta=\emptyset$ is genuinely needed and $\alpha>\beta$ does not suffice; or, independently, find a finitely generated nilpotent group $G$ and a positive-measure set $Y$ in a $G$-system with $\liminf_N \frac{1}{|F_N|}\sum_{g\in F_N}\mu(Y\cap T_g^2Y)=0$ for some F\u00f8lner sequence, which would refute Theorem 1.27. On the combinatorial side, a countable amenable group $G$ and a set $A\subseteq G\times G$ with $d^*_l(A)>0$ for which no $B$ with $d_F(B)\ge d^*_l(A)^2$ satisfies $B\times B\subseteq AA^{-1}$ would refute Theorem 1.16.

Watch

Extended reading notes

Core claim

The central discovery is that sets of returns contain product sets in two distinct regimes. First, for any group $G$, any $G\times G$-system, and any positive-measure $Y$, the return set $\{(g,h):\mu(Y\cap T_{(g,h)}Y)>0\}$ contains $B\times B$ with $d_F(B)\ge \mu(Y)^2$; when $G$ is amenable this yields $B\times B\subseteq AA^{-1}$ for every $A\subseteq G\times G$ with $d^*_l(A)>0$ and every F\u00f8lner sequence $F$, with $d_F(B)\ge d^*_l(A)^2$. Second, the paper shows when the same holds for the product $BB$ inside return sets in $G$ itself: this is controlled by the symmetric correlation function $g\mapsto\mu(T_g^{-1}Y\cap T_gY)$, and for finitely generated nilpotent groups the averages of these correlations are uniformly bounded below by $\lambda_G\mu(Y)^2$ with $\lambda_G=2^{-n(n-1)/2}$, where $n$ is the size of a unipotent upper-triangular matrix group of which $G$ is a homomorphic image. Polynomial analogues assert that finite unions of products $p_j(B)q_j(B)$ of polynomial images lie in $AA^{-1}$ for nilpotent targets.

Load-bearing premise

The polynomial results for nilpotent groups rest on the assertion that Zorin-Kranich's nilpotent IP-polynomial multiple recurrence theorem remains valid when the derivative condition is switched from $\alpha\cap\beta=\emptyset$ to $\alpha>\beta$; Section 5.1 gives a case-by-case audit, but if any of those checks is wrong, Theorem 5.7 and all applications in Section 5 fail.

Editorial extensions

If this is right

  • For every countable amenable group $G$, every $A\subseteq G\times G$ with $d^*_l(A)>0$ has $AA^{-1}$ containing a Cartesian square $B\times B$ with $d_F(B)\ge d^*_l(A)^2$ for any prescribed F\u00f8lner sequence $F$; the same holds for right density and $A^{-1}A$ separately, and for two-sided versions in which both left and right densities are used.
  • For countable abelian groups, every positive-density $A\subseteq G$ has $B+B\subseteq A-A$ with $d_F(B)\ge d^*(A)^2$, recovering and sharpening classical results of Bergelson for $\mathbb{Z}$.
  • For finitely generated nilpotent groups, the symmetric ergodic average $\lim_{N}\frac{1}{|F_N|}\sum_{g\in F_N}\mu(Y\cap T_g^2Y)$ is uniformly bounded below by $\lambda_G\mu(Y)^2$, so every positive-density $A\subseteq G$ contains $B$ with $BB\subseteq AA^{-1}$ and positive density along any F\u00f8lner sequence.
  • Semidirect products of abelian groups, upper-triangular matrix groups $UT_n(R)$ over countable commutative rings, and $GL_n(Q)$ for $Q$ a countable algebraic extension of a finite field all have the symmetric averaging recurrence (SAR) property, implying $BB\subseteq AA^{-1}$ with explicit density bounds.
  • Polynomial versions hold for finitely generated $G$ and nilpotent $H$: for any polynomials $p_j,q_j:G\to H$ vanishing at the identity, $\bigcup_j p_j(B)q_j(B)\subseteq AA^{-1}$ for some positive-density $B\subseteq G$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The explicit constant $\lambda_G=2^{-n(n-1)/2}$ is an artifact of the proof's choice of a finite-index subgroup of $UT_n(\mathbb{Z})$; a different embedding or a more economical subgroup construction would likely improve the bound substantially, so the true nilpotent constant is probably much larger.
  • The general probability-measure formulation (Theorem 4.2) suggests that the product-set-in-returns phenomenon is not tied to F\u00f8lner sequences but holds along any sequence of probability measures that is asymptotically invariant, which opens the door to random-walk averages on non-amenable groups where no F\u00f8lner sequence exists.
  • The sharp contrast between $G\times G$ (Cartesian squares always appear) and $G$ itself (products only for certain amenable classes) points toward a possible characterization: the SAR property may fail exactly when a group admits a 'squaring collapse' analogous to the quaternion example in Remark 1.26, suggesting a testable dichotomy for amenable groups.
  • The polynomial results for nilpotent targets are proved via an IP-polynomial multiple recurrence theorem; if the reconciliation in Section 5.1 is correct, a similar route could yield polynomial versions for averages along sparse sequences like $\{p_n^c\}$ in nilpotent groups, extending Corollary 1.8 beyond $\mathbb{Z}$.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The paper studies sets of measurable returns in countable groups and product sets inside sets of differences. Its central theorem (Theorem 1.16, with ergodic counterpart Corollary 1.15) states that for a countable amenable group G, any A ⊆ G×G with positive left upper Banach density contains a subset B×B with d_F(B) ≥ d*_l(A)^2 for every prescribed left or right Følner sequence F. This is derived from an abstract product-in-returns theorem (Theorem 3.1) via an intersectivity lemma. The paper also studies non-Cartesian analogues BB ⊆ AA^{-1}: for finitely generated nilpotent groups it proves a quantitative lower bound for symmetric ergodic averages (Theorem 1.27), and it establishes the SAR property for further classes (semidirect products of abelian groups, UT_n(R), and GL_n over unions of finite fields). Section 5 contains polynomial multiple-recurrence results (Theorems 1.46 and 1.48) based on a modification of Zorin-Kranich's nilpotent IP-polynomial theorem. Section 7 gives counterexamples showing subtle left/right asymmetric behaviour in the Heisenberg group.

Significance. If the results hold, this is a substantial and quantitative generalization of Bergelson's classical difference-set theorems: it passes from Z and Z^2 to arbitrary countable amenable groups, works with arbitrary Følner sequences, and gives explicit lower bounds. The proof architecture around Theorem 3.1 is clean, and the applications to free groups via Guivarc'h's theorem and the explicit counterexamples in Section 7 are valuable. The main weakness is that the polynomial part of the paper depends on a modified reading of a deep external theorem, and the manuscript's audit of that modification is not a complete proof; the same applies to a technical lemma in the construction of FVIP systems. With those points repaired, this would be a strong contribution.

major comments (4)
  1. [§5.1, Remark 5.5 and Theorem 5.7] The paper changes the defining condition in Equation (40) from α∩β=∅ to α>β and then invokes [ZK14, Theorem 5.32] to obtain Theorem 5.7. The verification in the paragraph beginning 'Checking the results in [ZK14] with the condition α>β' is a list of assertions ('seems to use our definition', 'still work') rather than a proof that the entire argument of [ZK14, Theorem 5.32], including the group property of IP-G•-polynomials, the FVIP closure axioms, and the IP-limit step in [ZK14, Lemma 5.7], remains valid under the weakened definition. Since Theorems 1.46 and 1.48 depend on Theorem 5.7, this is load-bearing: the authors should either give a complete proof of the modified theorem, identify a published source for it, or state Theorem 1.48 with the exact external hypothesis made explicit.
  2. [§5.2, Lemma 5.10] The proof that P0(g,F•) is an FVIP group asserts the identity D_{gh}q = D_gq · D_hq · D_gD_hq. This identity is not valid as written for polynomial maps into non-abelian nilpotent groups; for example, q(x)=(0,x,x^2) in UT_3(Z) with G=Z does not satisfy it. Since Lemma 5.10 is needed for Corollary 5.12 and hence for Theorem 1.48, the closure argument must either be corrected or replaced by a reference.
  3. [§4, proof of Theorem 4.12] Corollary 4.11 gives a lower bound for the Cesàro averages C_N = (1/N)∑_{n=1}^N a_n, not for the sequence a_N itself. The displayed chain 'lim_N a_N ≥ lim_N C_N' is therefore unjustified. If the density d_{(ν_N)} is the upper density (limsup), the proof should apply Corollary 4.4 with λ = limsup a_N; as written, the proof of Theorem 4.12 is invalid. This does not affect the main amenable-group theorem, but it is load-bearing for the stated general non-amenable result.
  4. [§2, Lemma 2.2] The proof defines f as the pointwise limit of the averages (1/|F_N|)∑_{g∈F_N} 1_{Y_g} and applies Fatou's lemma. For arbitrary sequences F_N and arbitrary measurable sets Y_g there is no reason for the pointwise limit to exist. The proof can be repaired by taking a limsup and using a subsequence attaining the upper density, but the current argument should be rewritten with explicit limsup notation and a justification of the Fatou-type inequality used.
minor comments (4)
  1. [§5.1, Proposition 5.4] The displayed composition 'α^{-1}∘ψ∘β' appears to be a typo; it should read 'α^{-1}∘φ∘β'.
  2. [§1.2, Corollary 1.10] The proof of Theorem 1.9 gives a limit along the even balls B_{2N}, while Corollary 1.10 states a density along the full sequence (B_N). A sentence explaining why the limsup along B_N is bounded below by the limit along B_{2N} would remove ambiguity.
  3. [§4, Theorem 4.2] The approximation of a general sequence (ν_N) of probability measures by empirical measures on a countable set is asserted without proof or reference; a brief argument or citation would make this step easier to check.
  4. [Introduction] There are several typographical issues, for example 'pertaning' in the first paragraph of Section 1 and inconsistent notation for upper densities with missing overlines in several displayed formulas.

Circularity Check

0 steps flagged · score 0.0 of 10

The derivation chain is self-contained with respect to its inputs; no step reduces a claimed prediction to a fitted parameter or to a self-citation.

full rationale

I walked the main derivation chains. The combinatorial Theorem 1.16 is obtained from Corollary 1.15, which follows from Theorem 3.1/Corollary 3.2 and Proposition 1.14 (the amenable-group von Neumann mean ergodic theorem, proved at the start of Section 2). Theorem 3.1 is proved directly from the intersectivity lemma (Lemma 2.2): the same averaged quantity λ appears in the hypothesis and in the lower bound for d_F(B), but this is a transfer construction, not an assumption of the conclusion. The passage from measurable returns to difference sets uses Lemma 2.7, which invokes Furstenberg's correspondence principle from [RM25, Theorem 3.14]. This is a self-citation, but the cited theorem is a standard, parameter-free correspondence between densities and measure-preserving systems; it does not incorporate the target result (B×B or BB inclusions) and therefore is independent support under rule 4. For Theorem 1.27 the proof embeds the nilpotent group into UT_n(Z), builds explicit Følner sets A_N, B_N, and invokes Malcev's embedding, Lemma 5.15, and Zorin-Kranich's [ZK16, Theorem 1.1]; no fitted constants or target inequalities are imported. The polynomial theorems 1.46 and 1.48 rest on [ZK14, Theorem 5.32] with the IP-polynomial derivative condition weakened from α∩β=∅ to α>β. The paper's Section 5.1 audit ('Checking the results in [ZK14] with the condition α>β') is a case-by-case verification and not machine-checked, so this is a genuine external-theorem correctness risk; but it is not circular, because the quoted result is not the theorem being proved and no step reduces to its own input. No fitted-parameter predictions, no author-imported uniqueness claims, and no ansatz smuggled in via citation were found. The only mild ground for a non-zero score is the [RM25] self-citation for a classical correspondence lemma; it is not load-bearing, and the core derivation stands on independent ergodic facts.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claims are universal over all measure-preserving systems; no constants are fitted to data. The paper's concrete contribution is the proof architecture: an intersectivity lemma, the SAR property, and transfer via Furstenberg correspondence. The heavy lifting is outsourced to standard ergodic theorems (von Neumann, ZK14, ZK16, Leibman, Malcev), all external. No invented entities or free parameters.

assumptions (6)
  • standard math Furstenberg correspondence principle for countable amenable groups (Lemma 2.4, cited to [RM25, Theorem 3.14])
    Bridges the combinatorial object A and the measure-preserving system (X,B,μ,(T_g)); indispensable for Theorems 1.16, 1.25, and 1.46.
  • standard math von Neumann mean ergodic theorem for amenable group actions (Proposition 1.14)
    Yields lim inf (1/|F_N|) ∑ μ(Y∩T_gY) ≥ μ(Y)^2, the engine of the Cartesian-square results.
  • standard math Zorin-Kranich [ZK16, Theorem 1.1]: Følner-independent limits of nilpotent ergodic averages
    Used as Theorem 1.29 to promote the lower bound from one Følner sequence to all Følner sequences in Theorem 1.27.
  • standard math Zorin-Kranich [ZK14, Theorem 5.32]: nilpotent IP-polynomial multiple recurrence
    Core input for Theorems 1.46 and 1.48; the paper modifies the IP-polynomial derivative condition and audits the proof, but does not reprove the theorem.
  • standard math Malcev embedding: torsion-free f.g. nilpotent groups embed in UT_n(Z)
    Replaces a general f.g. nilpotent G by a matrix group where explicit finite-index subgroups with controlled squaring map can be built.
  • standard math Leibman's polynomial mapping theory ([Lei02b])
    Defines polynomial maps between nilpotent groups and supplies closure properties used throughout Section 5.

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Pith. "Pith review of Product sets in sets of returns and positivity of symmetric ergodic averages." pith.science (2026). https://pith.science/paper/AQ3CSZC7

@misc{pith2026260802873,
  author       = {Pith},
  title        = {Pith review of: Product sets in sets of returns and positivity of symmetric ergodic averages},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AQ3CSZC7}},
  note         = {Machine review of arXiv:2608.02873}
}
abstract

We study sets of (measurable) returns in countable groups $G$, namely sets of the form $\{g\in G:\mu(A\cap T_gA)>0\}$ arising from measure-preserving actions. Extending a result of Bergelson, we show that sets of returns in $G\times G$ contain subsets of the form $B\times B$, where $B$ is large with respect to suitable notions of largeness that remain meaningful even for non-amenable groups. As a consequence, if $G$ is amenable, then every sufficiently large subset $A\subseteq G\times G$ satisfies $B\times B\subseteq AA^{-1}$ for some large set $B\subseteq G$. We also investigate when sets of returns in $G$ contain product sets $BB$ with $B$ large. In contrast with the Cartesian-product phenomenon above, this problem is considerably subtler in non-abelian groups and is closely connected to `symmetric correlation functions', namely functions of the form $g\mapsto \mu(T_g^{-1}A\cap T_gA)$. We use this connection to show that, for broad classes of amenable groups - including finitely generated nilpotent groups and certain solvable non-nilpotent groups, every sufficiently large set $A\subseteq G$ contains a large subset $B$ satisfying $BB\subseteq AA^{-1}$. Finally, we establish polynomial analogues of these results for finitely generated nilpotent groups, extending earlier work of Bergelson and Ruzsa.

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