{"id":"ec02d357-57d2-43f4-927d-62b370728a20","arxiv_id":"2506.17771","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"For rational-spectrum IP sequences, multiple ergodic averages along finite-sum sets are claimed to converge to nilsystem limits with positive-IP-density large intersections, though a central proof step is flawed as written.","lead":"This paper studies averages over IP sets, the collection of all finite sums of a chosen infinite sequence of times, and gives convergence criteria for them. For a special rational spectrum class it claims large-intersection estimates for three- and four-term arithmetic patterns, but the proof has a gap in a key lemma.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 5.4's weight-absorbing identity drops the factor η(πx)^{s+t}; Theorems 6.2, 6.4, and Theorem 1.1 therefore lack a valid proof as written.","rationale":"I read the full preprint in good faith. Sections 3 and 4 contain plausible IP analogues of spectral criteria, van der Corput, seminorms, and characteristic factors; the pointwise nilsystem results in Section 5 are substantial. However, the central application to large intersections depends on Lemma 5.4, whose key algebraic identity is false as written. The reader's weakest_assumption identifies exactly this step, and my own expansion confirms it: for a character η, the claimed absorption of η(na) into f'_1,f'_2 produces an extra factor η(πx)^{s+t}; this factor is nontrivial for most coprime pairs, since s+t=0 only when |ℓ1−ℓ2|=1. Because Lemma 5.4 feeds directly into Theorems 6.2 and 6.4, and through them into Theorem 1.1, the central claim is not established by the manuscript. I also note an additional inconsistency that reinforces the reader's verdict: Theorem 6.2 states that Lemma 5.4 produces a 1-step nilsystem factor, whereas Lemma 5.4 with k=3 produces a (k−1)=2-step factor; the subsequent reduction to T^m×D is therefore not justified as written. This is not a disagreement with the consensus that large intersections hold in ordinary ergodic theory; it is a concrete internal gap in the proof. No amount of external plausibility repairs the missing factor, and no machine-checked verification is claimed. A direct algebraic counterexample to the equality settles the matter independently of any choices made later in the proof.","tokens_in":31435,"tokens_out":17486,"duration_ms":180339,"concrete_test":"Work in the setting of Lemma 5.4 with X=Z1(X)=T, T(x)=x+α for irrational α, ℓ1=1, ℓ2=2, Bezout coefficients s=1,t=0, η(x)=e^{2π i x}, and f1=f2≡1. The asserted equality becomes E_{n∈IP} η(nα) = E_{n∈IP} η(x+nα) for every x, i.e. η(nα) is claimed equal to η(x)η(nα) for each n. Evaluating at any n with η(nα)≠0 shows the two sides differ by η(x), which is not 1 for almost every x. This directly disproves the first displayed identity in Lemma 5.4 for a nonconstant character, without any approximation or limiting argument.","verdict_should_be":"REJECT","load_bearing_attack":"Lemma 5.4 is the bridge from Host–Kra characteristic factors to the limit formulae used for the large intersection theorems. In the first reduction, the proof takes η a character, chooses s,t with sℓ1+tℓ2=1, and sets f'_1=f_1η^s, f'_2=f_2η^t, then asserts E_{IP} η(na) ∏ f_i(T^{ℓ_i n}x) = E_{IP} ∏ f'_i(T^{ℓ_i n}x). Expanding on the Kronecker factor Z1 (rotation by a, factor map π), the right-hand side contains η(πx+ℓ1na)^s η(πx+ℓ2na)^t. For a character η this equals η(πx)^{s+t} η(na), while the left-hand side is η(na). The two expressions differ by the factor η(πx)^{s+t}, which is not identically 1 for general coprime ℓ1,ℓ2 unless s+t=0, i.e. unless ℓ1−ℓ2=±1. Thus the identity is false already for k=2 with f1=f2=1. Since Theorems 6.2 and 6.4 invoke Lemma 5.4 to replace 1_A by its nilfactor projection and to pass to Corollary 5.8, the main combinatorial claim Theorem 1.1 is not proved as written. The paper contains substantial independent material in Sections 3–5, but the large-intersection argument depends on this unsound step.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript develops a theory of multiple ergodic averages sampled along IP sets generated by a sequence (n_j). Section 3 gives spectral criteria for mean convergence and defines the notion of rational spectrum, with a limit formula on the rational Kronecker factor. Section 4 introduces IP versions of the van der Corput lemma, Gowers-Host-Kra seminorms, and cubic measures, and proves that the Host-Kra factors are characteristic for IP averages when the generating sequence has rational spectrum. Section 5 establishes limit formulas in nilsystems, including pointwise convergence for systems with disjoint spectrum and weighted limit formulas on synchronized nilsystems. Section 6 uses these formulas to prove positive IP-density of large intersections for three- and four-term configurations, yielding the headline Theorem 1.1 for sets of positive upper Banach density when the generating sequence has rational spectrum. Appendix B gives a counterexample showing that the large-intersection conclusion fails for arbitrary sequences.","tokens_in":31733,"tokens_out":9771,"duration_ms":99487,"significance":"If the main theorem were correct, it would provide an IP analogue of the Bergelson-Host-Kra large-intersection theorem and would be a substantial contribution to the ergodic theory of IP convergence. The paper contains several independent contributions that appear sound: the spectral characterization in Theorem 3.3, the rational-spectrum limit formula in Corollary 3.8, the IP van der Corput lemma, and the construction of rational IP cubic measures in Proposition 4.7. However, the route from the nilsystem limit formulas to the large-intersection theorems passes through Lemma 5.4, whose main reduction is algebraically invalid. As a result, Theorems 6.2, 6.4, and Theorem 1.1 are not proved as written; the significance of the paper is therefore conditional on a repair of Section 5.2.","major_comments":[{"comment":"The identity used to absorb the weight η into f1 and f2 is false. For a character η of Z1 with factor map π:X→Z1, after choosing integers s,t with sℓ1+tℓ2=1 and setting f1'=f1 η^s and f2'=f2 η^t, the product on the right contains η(πx+ℓ1na)^s η(πx+ℓ2na)^t, which equals η(πx)^(s+t) η(na). The left-hand side contains only η(na), so the asserted equality of averages omits the factor η(πx)^(s+t). This factor is not identically 1 for arbitrary coprime ℓ1,ℓ2; it is identically 1 only when s+t=0, i.e. when |ℓ1−ℓ2|=1. The claimed equality of the two averages fails already for k=2 with f1=f2=1. Since Lemma 5.4 is the bridge from the Host-Kra characteristic-factor theorem to the weighted nilsystem formulas used in Theorems 6.2 and 6.4, the large-intersection proof does not go through as written.","section":"§5.2, Lemma 5.4"},{"comment":"The same proof contains an incorrect orthogonality assertion: it claims that f_i' is orthogonal to Z_{k-1}(X) if and only if f_i is orthogonal to Z1(X). Because η is measurable with respect to Z1 and Z1 is a subfactor of Z_{k-1}, the conditional expectation satisfies E(f_i' | Z_{k-1}) = η^s E(f_i | Z_{k-1}); hence orthogonality of f_i' to Z_{k-1} is equivalent to orthogonality of f_i to Z_{k-1}, not to Z1. This matters because the subsequent application of Theorem 4.2 requires vanishing conditional expectation on Z_{k-1}, not merely on Z1. Even if the algebraic identity in the previous comment were repaired, this step would need a different justification.","section":"§5.2, Lemma 5.4"},{"comment":"The first sentence of the proof states 'we can absorb η into f1,f2 and so without loss of generality we assume that η=1'. This is precisely the same invalid reduction as in Lemma 5.4 and no alternative argument is supplied. Corollary 5.8 is the weighted nilsystem limit formula invoked in the proofs of Theorems 6.2 and 6.4, so the error is not confined to an auxiliary lemma; it directly affects the derivation of the main combinatorial claims.","section":"§5.2, Corollary 5.8"}],"minor_comments":[{"comment":"The heading contains a typo: 'Exmaple 2.4' should be 'Example 2.4'.","section":"Page 3, first paragraph of §1.1"},{"comment":"The notation σ((n_j)_{k∈N}) uses the wrong index; it should be σ((n_j)_{j∈N}).","section":"Definition 3.7"},{"comment":"The lemma assumes η:Z1→R is nonnegative and real-valued, but the proof first treats η as a character, which is complex-valued. This mismatch should be clarified even apart from the algebraic error discussed above.","section":"Lemma 5.4 statement"},{"comment":"Near the end of the proof there is an extraneous 'Th' after 'for all rational s∈S1'; it should be removed.","section":"Proposition 5.7 proof"},{"comment":"The definition of positive lower IP-density is given with respect to every increasing Følner sequence, but the proof constructs a set using a single union of Følner blocks. The intended notion of density should be stated more explicitly so that the example satisfies the definition.","section":"Appendix A, Lemma A.1"},{"comment":"The citation 'a theorem of Leibman [6, Theorem 2]' is too imprecise for the reader to verify the claimed equivalence between vanishing of the U^k norm and vanishing of all Host-Kra integrals; a precise statement or a precise reference would be helpful.","section":"Lemma 4.10"}],"recommendation":"reject","confidential_remarks":"The paper contains substantial original material in Sections 3–5, and the main theorem might be salvageable with a fundamentally different weighted-average argument. However, the central proof chain as written is invalid at a load-bearing point, and the error is not a local typo: Lemma 5.4's reduction is algebraically false for general coprime coefficients, and the same flawed reduction is reused in Corollary 5.8. I recommend rejection rather than major revision because the required repair appears to require reworking the weighted limit formulas in Section 5.2 and the applications in Section 6."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague — quick read of Kra–Shalom, arXiv:2506.17771. The reject verdict is right, but not because the paper is thin: the IP machinery is new and substantial, and the convergence criteria look healthy. The problem is a load-bearing false identity in Lemma 5.4. The stress-test is accurate. After choosing s,t with s·ℓ1 + t·ℓ2 = 1 and absorbing η^s, η^t into f1, f2, the asserted equality E_IP η(na) ∏ f_i(T^{ℓ_i n}x) = E_IP ∏ f'_i(T^{ℓ_i n}x) expands on the Kronecker factor to differ by η(πx)^{s+t}. You only get the left side if that factor is 1, i.e. s+t = 0, which is not generally true for coprime ℓ1,ℓ2. So Lemma 5.4 fails as stated, and since it is the bridge to Theorems 6.2, 6.4 and Theorem 1.1, the large-intersection claims are not proved as written.\n\nCredit where due: Section 3's spectral criteria (Theorem 3.3) and the infinite-product rewriting are clean and correct as far as I can see. The IP van der Corput lemma, the IP Gowers–Host–Kra seminorms and cubic measures in Section 4 are genuinely new tools, and the characteristic factor theorem (Theorem 4.2) may well be salvageable. The nilsystem limit formula in Theorem 5.2 is a real extension of earlier work by Ziegler and by Bergelson–Host–Kra. The paper is not careless overall; the remainder is careful, and the flaw is a localized algebraic slip in one reduction. That said, it sits exactly where the main application depends.\n\nOne more minor point: Lemma 5.4 states η is real-valued, but the proof begins \"We start with the case that η is a character\" — a sign of sloppiness. The appendix on large-intersection failure is a nice complement, but it does not patch the gap.\n\nWho is this for? Ergodic theorists and additive combinatorists working on IP recurrence. The machinery in Sections 3–4 will be cited if fixed, and even now the paper is worth a serious referee. But I would not cite the main result until Lemma 5.4 is repaired. My recommendation: send it to peer review and ask the authors to fix Lemma 5.4. If the fix is straightforward, the paper could become correct; as written, it should not be accepted.","headline":"The IP machinery is real and interesting, but a false identity in Lemma 5.4 undercuts the paper's main large-intersection theorem.","tokens_in":32261,"tokens_out":3095,"would_cite":false,"duration_ms":31067,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37A15","11B30","28D15","37A35"],"pacs":[],"model":"deepseek-v4-flash","headline":"IP averages with rational spectrum are controlled by nilsystems and force large intersections of size d(E)^3 and d(E)^4.","keywords":["multiple ergodic average","IP-set","nilsystem","rational spectrum","large intersection property","Host-Kra factor","Følner sequence","IP convergence"],"falsifier":"Expanding the right-hand side of the identity in Lemma 5.4 for $s=1$, $t=0$, $\\ell_1=1$, $\\ell_2=2$ gives an extra factor $\\eta(\\pi x)$ that is absent on the left; substituting $f_1=f_2=1$ and any allowed nonconstant weight $\\eta$ makes the two sides visibly different whenever $\\eta(\\pi x)\\ne 1$ on a set of positive measure. A direct computation of both sides in this configuration—for example on a rotation with $\\eta$ a nonconstant character and the weight normalized so the left side has limit $1$—decides whether the reduction is valid.","tokens_in":31215,"feed_emoji":"🔁","tokens_out":13932,"duration_ms":147827,"temperature":0.7,"pith_summary":"The paper develops convergence theory for ergodic averages taken over IP sets—all finite sums of a fixed sequence $(n_j)$—and uses it to prove large-intersection results along such sets. The main structural claim is that when $(n_j)$ has rational spectrum, the same Host–Kra factors and nilmanifold limits that govern ordinary multiple averages also govern IP averages. From this, the paper derives limit formulas for multiple weighted averages and, as a consequence, positive-IP-density lower bounds for three- and four-term correlations: for any set $E$ of positive upper Banach density, the overlaps exceed $d^*(E)^3-\\varepsilon$ and $d^*(E)^4-\\varepsilon$ on a positive-density subset of the IP. A separate example shows that without the rational-spectrum assumption, large intersections can fail even for very strong notions of largeness.","feed_headline":"IP averages with rational spectrum force large intersections","feed_subtitle":"If the averaging sequence's spectrum is rational, three- and four-term overlaps beat d(E)^3 and d(E)^4 on a positive-density IP subset.","key_machinery":"The engine is the rational IP Gowers–Host–Kra seminorm $\\|\\cdot\\|_{U^k_{\\mathrm{IP}}}$, built from relatively independent joinings over the rational Kronecker factor $K_{\\mathrm{rat}}(X)$ instead of the invariant factor, together with the associated IP cubic measures $\\tilde{\\mu}^{[k]}$. An IP version of the van der Corput lemma (Lemma 4.3) transfers control of the multiple average to these seminorms; Proposition 4.9 bounds the average by the seminorms, and Lemma 4.10 identifies their kernel with the Host–Kra factor $Z_{k-1}(X)$, proving Theorem 4.2. The limit formulas then pass through nilsystems: Theorem 5.3 gives pointwise convergence for a dense class of functions when spectra are disjoint, and Corollary 5.8 supplies the weighted limit on synchronized nilsystems that Theorems 6.2 and 6.4 use.","core_discovery":"The central claim is that, for an IP sequence with rational spectrum, multiple ergodic averages along the IP behave like ordinary multiple averages: the Host–Kra factor $Z_{k-1}(X)$ is characteristic (Theorem 4.2), and the limit is an integral over a nilmanifold (Theorem 5.2). On the combinatorial side, this gives a large-intersection theorem: for a set $E\\subseteq\\mathbb{Z}$ of positive upper Banach density, the set of $n$ in the IP for which $d^*(E\\cap(E-\\ell_1 n)\\cap(E-\\ell_2 n))>d^*(E)^3-\\varepsilon$ has positive IP-density with respect to every increasing Følner sequence, and the four-term analogue holds with $d^*(E)^4-\\varepsilon$ (Theorem 1.1). The proof builds IP versions of the van der Corput lemma, Gowers–Host–Kra seminorms, and cubic measures, and obtains pointwise convergence formulas on nilsystems for a dense class of functions. These are the IP counterparts of the classical large-intersection theorems for linear averages.","pith_inferences":["If the missing condition in Lemma 5.4 can be supplied—for instance by choosing $s,t$ with $s+t=0$ or by a separate argument for the extra factor $\\eta(\\pi x)^{s+t}$—then Theorems 6.2 and 6.4 follow as written; otherwise the proof of the large-intersection theorem needs a different bridge from the characteristic-factor theorem to the nilmanifold formula.","The same IP seminorm machinery suggests a polynomial IP large-intersection theorem: replacing $\\ell_i n$ by integer-valued polynomials should follow with the nilmanifold factor dimension raised accordingly.","The optimal constant in the non-IP three-term bound (Question 1.5) is plausibly tied to Behrend-type constructions, since the appendix's counterexample uses Behrend's sets with no three-term progressions; quantifying that connection could decide whether $C(\\delta)$ is exponential or subexponential.","The pointwise convergence question (Question 1.6) could be attacked by showing that the dense class in Theorem 5.3 is a convergence-determining class, so that pointwise limits on that class pass to all of $L^2$."],"forward_implications":["For base-$a$ digit sequences such as $n_j=10^{j-1}$, the IP averages exist in $L^2$ for every function, and under total ergodicity they converge to the projection onto the invariant functions, giving equidistribution of the IP along toral rotations.","The integer form of the large-intersection theorem holds for all sequences with rational spectrum, so in such IPs every positive-density set contains many arithmetic progressions of length 3 and 4 whose common difference lies in the IP.","Non-rational IPs can destroy large intersections: the appendix constructs an ergodic system and a positive-measure set for which every central set contains a central subset with no three-term large intersection, showing that the rational-spectrum condition is close to optimal.","The weighted nilmanifold limit (Corollary 5.8) gives a computable formula for the asymptotic size of correlations even when the Kronecker part is not disjoint from the sequence's spectrum.","The paper leaves open whether the large-intersection sets are syndetic and whether any non-rational IP can support the same conclusion, so the rational-spectrum condition is a natural boundary of the method."],"supporting_citations":[{"why":"Supplies the Host–Kra factors, the Gowers–Host–Kra seminorms, and the nilmanifold limit structure that the IP variants adapt and generalize.","marker":"[31]"},{"why":"Is the linear large-intersection theorem whose IP analogue the paper proves, and supplies the correspondence principle used to transfer the ergodic result to the integers.","marker":"[8]"},{"why":"Is the Furstenberg–Katznelson IP Szemerédi theorem, the baseline IP recurrence result that motivates the IP-Følner setup and the large-intersection question.","marker":"[24]"},{"why":"Provides the universal characteristic factor and the limit formula for ordinary multiple averages that the paper extends to IP averages.","marker":"[43]"},{"why":"Gives the theorem used to replace measurable cocycles by continuous ones, which is needed in the pointwise convergence arguments for nilsystems.","marker":"[36]"},{"why":"Supplies the four-term lower bound in 2-step nilsystems used in the proof of Lemma 6.3, which Theorem 6.4 invokes.","marker":"[17]"},{"why":"Defines the closed subgroup $\\tilde G$ whose compact quotient carries the nilmanifold integral in Theorem 5.2.","marker":"[35]"}],"fun_headline_variants":["IP averages with rational spectrum force large intersections","Rational-spectrum IP sets yield optimal overlap bounds","IP ergodic averages: rational spectrum implies large intersections","For rational IP sequences, averages converge like nilsystems","IP averages with rational spectrum: large intersections guaranteed"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise of the large-intersection proof is that Lemma 5.4's weighted-average reduction holds, which in the written argument requires the hidden condition $s+t=0$; the paper does not establish that condition.","fun_headline_variants_meta":{"raw":{"variants":["IP averages with rational spectrum force large intersections","Rational-spectrum IP sets yield optimal overlap bounds","IP ergodic averages: rational spectrum implies large intersections","For rational IP sequences, averages converge like nilsystems","IP averages with rational spectrum: large intersections guaranteed"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00045,"raw_usage":{"total_tokens":2244,"prompt_tokens":896,"completion_tokens":1348,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":512,"completion_tokens_details":{"reasoning_tokens":1275}},"tokens_in":512,"tokens_out":1348,"duration_ms":10218,"temperature":1.0,"reasoning_tokens":1275,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T19:04:23.239774+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Expanding the right-hand side of the identity in Lemma 5.4 for $s=1$, $t=0$, $\\ell_1=1$, $\\ell_2=2$ gives an extra factor $\\eta(\\pi x)$ that is absent on the left; substituting $f_1=f_2=1$ and any allowed nonconstant weight $\\eta$ makes the two sides visibly different whenever $\\eta(\\pi x)\\ne 1$ on a set of positive measure. A direct computation of both sides in this configuration—for example on a rotation with $\\eta$ a nonconstant character and the weight normalized so the left side has limit $1$—decides whether the reduction is valid.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the universal characteristic factor and the limit formula for ordinary multiple averages that the paper extends to IP averages."},{"cited_title":"Lindenstrauss","cited_arxiv_id":null,"evidence_quote":"Gives the theorem used to replace measurable cocycles by continuous ones, which is needed in the pointwise convergence arguments for nilsystems."},{"cited_title":"Frantzikinakis","cited_arxiv_id":null,"evidence_quote":"Supplies the four-term lower bound in 2-step nilsystems used in the proof of Lemma 6.3, which Theorem 6.4 invokes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the closed subgroup $\\tilde G$ whose compact quotient carries the nilmanifold integral in Theorem 5.2."}],"review_version":1}