arXiv:2608.02873v1

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

Vitaly Bergelson, Saúl Rodríguez-Martín

math.DSmath.COmath.GR37A1543A07

Paper metadata

Original submission: August 3, 2026
Claimed MSC: 37A15, 43A07
arXiv categories: math.DS, math.CO, math.GR
Paper license: arXiv non-exclusive distribution license ↗
Comments: 57 pages

Abstract

We study sets of (measurable) returns in countable groups $G$ , namely sets of the form $\{g\in G:μ(A\cap T_gA)>0\}$ 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 $g\mapsto μ(T_g^{-1}A\cap T_gA)$ . 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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Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

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Detailed mathematical audit

Generated August 18, 2026

01Statements5 reported findingsContains unsupported statements

The general product-set construction, its Cartesian and polynomial consequences, the semidirect-product and finite-field matrix results, and the principal Heisenberg counterexamples are supported. The quantitative symmetric-averaging bound for every finitely generated nilpotent group in Theorem 1.27 is not verified: its proof identifies the squares of one Følner sequence with another by a density assertion that fails for subgroups of the unitriangular group. The paper's qualitative nilpotent product-set conclusion remains correct through the independent polynomial-recurrence proof.

Theorems 3.1, 1.5, and 1.16–1.25Correct

Product and Cartesian-product sets in measurable and combinatorial return sets

Pages 3–9 and 16–29 · Sections 2–4 · arXiv:2608.02873v1

The intersectivity lemma selects a set $B$ of the claimed upper density for which every finite intersection of the associated measurable sets has positive measure. Applying it to $Y_g=\bigcap_i T_{\phi_i(g)}Y\cap T_{\psi_i(g)}^{-1}Y$ gives $\psi_i(B)\phi_j(B)\subseteq R_\mu^T(Y)$ by one action identity. The amenable-group, Reiter-sequence, logarithmic-density, free-group, and convolution-power corollaries then follow from the appropriate mean ergodic theorem and the stated correspondence principle. The left/right order of every difference set is preserved in these deductions.

Theorems 1.46 and 1.48Correct

Polynomial return sets and polynomial product-set consequences

Pages 14–15 and 30–39 · Section 5.1 and Section 5.2 · arXiv:2608.02873v1

Lemma 5.10 places the polynomial sequences generated by any IP-system inside a finitely generated VIP group. The nilpotent IP-polynomial recurrence theorem therefore makes the multiple-return set IP-star. Zorin-Kranich's norm-convergence theorem gives a Følner-sequence-independent limit; Lemma 5.14 rules out zero by extracting an IP-system from any zero-average sequence. Theorem 3.1 then converts this positive multiple average into all product-set inclusions in Theorem 1.46, and the Cartesian version follows in $H\times H$ .

Zorin-Kranich, nilpotent IP polynomial multiple recurrence ↗

Theorem 1.27Not able to verify

Uniform quantitative positivity for finitely generated nilpotent groups

Pages 9 and 39–42 · Theorem 1.27 and its proof · arXiv:2608.02873v1

The theorem asserts a system-independent bound $\lim_N\frac1{|F_N|}\sum_{g\in F_N}\mu(Y\cap T_{g^2}Y)\geq\lambda_G\mu(Y)^2$ for every left or right Følner sequence, with the displayed explicit choice of $\lambda_G$ . The proof's decisive approximation of a Følner average over $B^G$ by squares of elements of $A^G$ is false under the only established assumption that $G$ is a subgroup of $\operatorname{UT}_n(\mathbb Z)$ ; Part 2 gives a concrete subgroup where the alleged relative symmetric difference tends to $1/2$ . No counterexample to the averaging theorem itself is obtained, and the earlier polynomial-recurrence theorem proves positivity without this uniform constant, but the manuscript supplies no independent proof of the quantitative statement.

Theorems 1.31, 1.33, and 1.34Correct

Symmetric averaging for semidirect products and matrix groups

Pages 10–11 and 43–47 · Section 6 · arXiv:2608.02873v1

For $H\rtimes K$ with $H$ abelian, the square of $kh$ has $H$ -component $\phi_k(h)\phi_{k^2}(h)$ , an endomorphism in $h$ . Applying the mean ergodic inequality first in $H$ and then the SAR hypothesis in $K$ , followed by a diagonal Følner construction, gives the squared lower bound. Iteration handles $\operatorname{UT}_n(R)$ . For $\operatorname{GL}_n(Q)$ over an algebraic extension of a finite field, the finite subgroups form a two-sided Følner sequence; regular split matrices occupy asymptotic proportion $1/n!$ and decompose into diagonal abelian subgroups, giving the claimed $\mu(Y)^2/n!$ bound.

Theorems 7.1 and 7.5Correct

Heisenberg counterexamples separating left and right product phenomena

Pages 48–52 · Section 7 · arXiv:2608.02873v1

The rapidly separated Heisenberg boxes form a left Følner sequence and their pairwise quotients fall into three mutually recognizable coordinate regimes. Poincaré recurrence supplies a common nonzero central shift inside two positive-left-density sets; comparing the original and shifted products contradicts each of the three quotient regimes. The paired-box construction in Theorem 7.5 similarly forces the second middle coordinate to be bounded once the first is fixed, contradicting positive upper Banach density. These arguments establish the two advertised counterexamples.

02Proofs4 reported findingsContains incorrect or incomplete proofs

The main intersectivity and polynomial-recurrence chains are correct. The proof of Theorem 1.27 contains a false asymptotic-density identity and therefore does not establish its quantitative nilpotent bound. A separate finite-set avoidance formula in Proposition 7.6 is also written in the wrong order, but it has a verified local repair and does not affect the principal Heisenberg counterexamples.

Proof of Theorem 1.27Incorrect as written

Injectivity of squaring does not make its image asymptotically all of $B^G$

Page 41 · display beginning $|\operatorname{Sq}(A^G_{N_k})\triangle B^G_{N_k+1}|$ · arXiv:2608.02873v1

From $\operatorname{Sq}(A_N^G)\subseteq B_{N+1}^G$ and injectivity, the proof writes $\frac{|\operatorname{Sq}(A_N^G)\triangle B_{N+1}^G|}{|B_{N+1}^G|}=\frac{|B_{N+1}^G|-|A_N^G|}{|B_{N+1}^G|}\longrightarrow0.$ The preceding choice of $N_k$ controls only the ratios $|A_{N_k+1}^G|/|A_{N_k}^G|$ and $|B_{N_k+1}^G|/|B_{N_k}^G|$ ; it gives no comparison between the $A$ - and $B$ -counts. For a concrete failure, take $n=2$ and $G=\{I+4mE_{12}:m\in\mathbb Z\}\leq\operatorname{UT}_2(\mathbb Z).$ Then $A_N^G$ consists of the even choices of the parameter $a$ , whereas every parameter $b$ occurs in $B_N^G$ , so $|A_N^G|/|B_{N+1}^G|\to1/2$ and the displayed symmetric-difference ratio tends to $1/2$ . Downstream dependency: the comparison with the mean ergodic average on $B^G$ , and hence the torsion-free case of Theorem 1.27 and its quotient step. Repair classification: No repair supplied. A repair would need a specially proved embedding or congruence subgroup on which the square image has full relative density, or a different averaging argument.

Theorem 1.48 and Theorem 1.46Correct and complete

The IP-star and positive-average argument closes the qualitative polynomial results

Pages 30–39 · Lemmas 5.10–5.14 and proofs of Theorems 1.48 and 1.46 · arXiv:2608.02873v1

The derivative identities in Lemma 5.10 keep the generated family inside a finitely generated nilpotent polynomial group, and the separated-index convention used for the IP derivatives is checked at every material use of the cited recurrence theorem. Left and right Følner limits agree by inversion and a two-sided Følner sequence. If the common limit were zero, Lemma 5.14 constructs an IP-system on which the multiple-intersection values converge to zero, contradicting the positive IP-limit on a sub-IP-ring. This also independently proves the qualitative finitely generated nilpotent conclusion obtained from Theorem 1.46 with $p=q=\operatorname{id}$ .

Proposition 7.6Incorrect as written

The recursive forbidden set does not imply the claimed cross-disjointness

Page 52 · final paragraph of the proof of Proposition 7.6 · arXiv:2608.02873v1

To ensure $F_Ng_N\cap F_Mg_Ma=\varnothing$ for every pair $M,N$ , the proof says that it suffices to choose $g_N\notin E_N\cup\bigcup_{n<N}F_ng_nF_N^{-1}.$ That set has the factors in the wrong order and omits the two translates by $a$ needed for the two orientations of a cross-pair. Repair classification: Verified repair. At stage $N$ , choose $g_N$ outside $E_N\cup\bigcup_{n<N}F_N^{-1}F_ng_n\{a,a^{-1}\}.$ The first set has zero upper Banach density and the second is finite, so such a choice exists; the two elementary rearrangements of a hypothetical intersection then give exactly the two forbidden translates. This proves $A\cap Aa=\varnothing$ and completes Proposition 7.6 without changing its statement.

Sections 2–4 and 6Correct and complete

Intersectivity, correspondence, and iterated averaging arguments

Pages 16–29 and 43–47 · Sections 2–4 and 6 · arXiv:2608.02873v1

Fatou's lemma supplies the density-preserving point in the intersectivity lemma, and the null exceptional sets are removed simultaneously because the index set is countable. Every later product inclusion is a direct substitution into Theorem 3.1. The semidirect-product Følner sequence is diagonalized against finitely many translations at each stage; Jensen's inequality gives the second SAR exponent. The finite-field matrix count uses disjoint regular split eigenspace decompositions and loses exactly the factor $n!$ . No additional unresolved hypothesis or case was found in these proof chains.

03Novelty0 reported findingsNo non-novelty findings

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Paper: arXiv:2608.02873v1
Authors listed: Vitaly Bergelson, Saúl Rodríguez-Martín
Audit date: August 18, 2026

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

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Product and Cartesian-product sets in measurable and combinatorial return sets

Polynomial return sets and polynomial product-set consequences

Uniform quantitative positivity for finitely generated nilpotent groups

Symmetric averaging for semidirect products and matrix groups

Heisenberg counterexamples separating left and right product phenomena

Injectivity of squaring does not make its image asymptotically all of BGB^GBG

The IP-star and positive-average argument closes the qualitative polynomial results

The recursive forbidden set does not imply the claimed cross-disjointness

Intersectivity, correspondence, and iterated averaging arguments

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Injectivity of squaring does not make its image asymptotically all of $B^G$