{"id":"ff437c43-757b-44e9-bf92-2a2d6aa0393c","arxiv_id":"2608.02873","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Large subsets of countable amenable groups contain square product patterns inside their difference sets, with new positive results for nilpotent and solvable matrix groups.","lead":"Sets of measurable returns in products G×G are shown to contain Cartesian squares B×B with B large, extending Bergelson's 1985 theorem from Z to arbitrary countable groups. For products B·B inside G itself, the paper proves positivity for nilpotent and several solvable matrix groups using a new symmetric averaging recurrence property.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Polynomial theorems rest on a nonstandard weakening of Zorin-Kranich's IP-polynomial definition; the audit in §5.1 is plausible but unverified.","rationale":"The main Cartesian and nilpotent-averaging results appear sound: Theorem 3.1 follows cleanly from Lemma 2.2, Corollary 1.15 from Proposition 1.14, and Theorem 1.27 from Malcev embedding plus Theorem 1.29 and finite-index transfer. The only serious soft spot is the polynomial section, exactly as the reader identified. I agree with CONDITIONAL: the external-theorem reconciliation should be independently verified before the polynomial claims are treated as settled. The undefined valuation v in Lemma 6.7 is a minor presentational gap in a side result, not a reason to change the verdict.","tokens_in":46554,"tokens_out":26167,"duration_ms":248099,"concrete_test":"Have the authors either (a) prove that the concrete FVIP systems in Lemma 5.10 satisfy ZK14's original α∩β=∅ derivative condition, in which case the nonstandard reading is unnecessary, or (b) produce a line-by-line verification of [ZK14, Prop. 2.22, Lem. 4.20, Lem. 4.27, Lem. 5.7, Thm. 5.18, Thm. 5.26] under the α>β definition, confirming that every derivative computation and IP-limit interchange is used only in the ordered regime. A failure at any listed item invalidates Theorems 5.7, 1.46, and 1.48; a successful verification settles the concern.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The weakest point is the unverified modification of Zorin-Kranich's IP-polynomial framework in §5.1. Remark 5.5 weakens the defining equation (40) for IP-G•-polynomials from α∩β=∅ to α>β, and Theorem 5.7 is then extracted from [ZK14, Thm 5.32]. The paper's audit ('Checking the results in [ZK14]...') is a case-by-case assertion that ZK14 only ever invokes derivatives in the ordered regime; it is not a proof and is not machine-checked. The consequence is transitive: Theorems 1.46 and 1.48, the polynomial multiple recurrence and product-set results, depend on Theorem 5.7. If any step in ZK14's proof (e.g. Lemma 4.20 or the IP-limit manipulations in Lemma 5.7) genuinely requires disjointness for unordered pairs, the polynomial part of the paper collapses. This is not an internal contradiction but a load-bearing external-theorem assumption.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":46696,"tokens_out":29051,"duration_ms":277259,"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":[{"comment":"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.","section":"§5.1, Remark 5.5 and Theorem 5.7"},{"comment":"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.","section":"§5.2, Lemma 5.10"},{"comment":"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.","section":"§4, proof of Theorem 4.12"},{"comment":"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.","section":"§2, Lemma 2.2"}],"minor_comments":[{"comment":"The displayed composition 'α^{-1}∘ψ∘β' appears to be a typo; it should read 'α^{-1}∘φ∘β'.","section":"§5.1, Proposition 5.4"},{"comment":"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.","section":"§1.2, Corollary 1.10"},{"comment":"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.","section":"§4, Theorem 4.2"},{"comment":"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.","section":"Introduction"}],"recommendation":"major_revision","confidential_remarks":"The main Cartesian-product theorem (Theorem 1.16) and the surrounding ergodic arguments appear sound and are the strongest part of the paper. The central risk is Section 5: the paper relies on a modified version of Zorin-Kranich's deep theorem, and the current audit is not a proof. I would recommend asking the authors to supply a complete verification of that modification, or to restructure the polynomial claims so the external theorem is used only in its published form. The derivative-closure gap in Lemma 5.10 should also be fixed; both issues are plausibly repairable but currently block acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe paper is a serious advance. It takes Bergelson's 1985 results on difference sets and returns for Z and Z^2 and extends them to all countable amenable groups, with explicit density bounds for the Cartesian product case. The main engine, Theorem 3.1 (via Corollary 3.2), is a clean generalization of the intersectivity lemma: from a lower bound on an average of intersections you get a set B with positive density along any prescribed Følner sequence and a product set inside the return set. That is a useful tool, and the paper applies it widely: Følner and Reiter sequences, logarithmic density, even non-amenable free groups through Guivarc'h's ergodic theorem. The combinatorial corollary that every positive upper Banach density A in G×G contains B×B with d_F(B) ≥ d*_l(A)^2 and B×B ⊆ AA^{-1} is a genuine strengthening of Bergelson.\n\nI also like the nilpotent section. The SAR property is a natural notion, and Theorem 1.27 gives an explicit constant λ_G = 2^{-n(n-1)/2} for finitely generated nilpotent groups. The proof via Malcev embedding and the cleverly chosen subgroups A and B is elegant. The semidirect product and GL_n over locally finite fields results extend the range considerably, and the counterexamples in the Heisenberg group that separate left and right upper Banach density are instructive.\n\nThe soft spot is exactly where the reader's report puts it: the polynomial results (Theorems 1.46 and 1.48) rest on Theorem 5.7, which is Zorin-Kranich's nilpotent IP-polynomial multiple recurrence theorem with the derivative condition changed from disjointness to an ordering condition. The authors provide a section-by-section audit of ZK14, arguing that only ordered derivatives are ever invoked. That audit is plausible and I have not found an obvious gap, but it is not a proof, and the consequence is transitive: if one step in ZK14 genuinely needs disjointness for unordered pairs, the polynomial theorems collapse. This is a localized concern, not a general one—the main Cartesian results and the nilpotent constant do not depend on it. For a paper this ambitious, however, the referee should push for either a complete proof of Theorem 5.7 or a more formal verification of the reconciliation.\n\nThe ChatGPT provenance note in Lemma 6.7 is odd but harmless; the definitions are explicit and the lemma is proved.\n\nWho should read this: anyone working on recurrence, difference sets, or multiple ergodic averages for amenable groups. It deserves a serious referee, not a desk reject. The referee's job is to verify the ZK14 dependency and check a few of the longer proofs. I would engage with it.","headline":"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.","tokens_in":47251,"tokens_out":2878,"would_cite":true,"duration_ms":25243,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37A15","37A30","05D10","11B05"],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["sets of returns","product sets","upper Banach density","amenable groups","nilpotent groups","symmetric ergodic averages","polynomial recurrence","Følner sequences"],"falsifier":"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.","tokens_in":46312,"feed_emoji":"🔁","tokens_out":5258,"duration_ms":46441,"temperature":0.7,"pith_summary":"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 $2^{-n(n-1)/2}\\mu(Y)^2$.","feed_headline":"Every large set in G×G hides a large square B×B","feed_subtitle":"For any Følner sequence, B has density at least A's squared density; nilpotent groups get explicit bounds on symmetric averages.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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 $\\mathbb{Z}$."],"forward_implications":["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 $B\\subseteq G$."],"supporting_citations":[{"why":"Provides the original theorems in $\\mathbb{Z}$ and $\\mathbb{Z}^2$ that this paper extends to general groups and F\\u00f8lner sequences.","marker":"[Ber85]"},{"why":"Supplies the earlier polynomial sumset-in-difference-set results that Theorems 1.46 and 1.48 generalize.","marker":"[BR09]"},{"why":"The nilpotent IP-polynomial multiple recurrence theorem that Theorem 5.7 and all Section 5 polynomial applications depend on.","marker":"[ZK14]"},{"why":"Establishes norm convergence of multiple ergodic averages on amenable groups, used to prove the existence and F\\u00f8lner-independence of the limit in Theorem 1.29 and Theorem 1.48.","marker":"[ZK16]"},{"why":"Provides the inverse Furstenberg correspondence principle used in Lemma 2.4 to translate between sets of differences and sets of returns.","marker":"[RM25]"},{"why":"Gives the lower bound for ergodic averages along powers used in Proposition 3.10 to obtain uniform return-set products without explicit F\\u00f8lner sequence constraints.","marker":"[Lei02a]"},{"why":"Defines polynomial mappings between groups, the framework in which the nilpotent polynomial results are formulated and proved.","marker":"[Lei02b]"},{"why":"Provides the convolution ergodic theorem for probability measures on countable groups, used in Theorem 4.9 and Corollary 4.11 for non-amenable groups.","marker":"[JRT94]"}],"fun_headline_variants":["Return sets hide product squares B×B universally","Symmetric averages force large product structure in returns","In all groups, return sets contain B×B with B large","From return sets to product sets: the role of symmetry","Product sets in return sets: from squares to BB"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Return sets hide product squares B×B universally","Symmetric averages force large product structure in returns","In all groups, return sets contain B×B with B large","From return sets to product sets: the role of symmetry","Product sets in return sets: from squares to BB"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001277,"raw_usage":{"total_tokens":5313,"prompt_tokens":1128,"completion_tokens":4185,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":744,"completion_tokens_details":{"reasoning_tokens":4107}},"tokens_in":744,"tokens_out":4185,"duration_ms":23155,"temperature":1.0,"reasoning_tokens":4107,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T14:57:09.995247+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}