{"id":"a1b27fd0-0f4e-4ffb-b911-7e01eb34b04d","arxiv_id":"2411.17448","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Every square-difference-free subset of {1,...,N} has size at most N exp(-c sqrt(log N)), for some absolute constant c.","lead":"The paper proves that any set of integers up to N with no two elements differing by a square has size at most about N times the exponential of a negative constant times the square root of the logarithm of N. This is a major improvement over the previous best bound and introduces a new technique based on hypercontractivity into additive number theory.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proof of Claim 4.2 contains a false bound (2^{k+1}≤4k) at a load-bearing step; the exact factor appears repairable, but the manuscript as written is not fully correct.","rationale":"The reader identified the external Theorem 2.5 as the weakest assumption. That is a legitimate risk, and the Appendix A adaptation is not fully formal, but my independent reading found a more concrete internal problem: the proof of Claim 4.2 uses a numerically false inequality, 2^{k+1}≤4k, at the heart of the induction establishing Theorem 1.2. This is a real flaw in the manuscript as written. However, the flaw appears repairable because the exact factor 2^{2k+1} is still dominated by C0^{k/2} = 8192^{k/2} = 2^{6.5k} for all k≥1, and the auxiliary factors (L/L')^{(d-k)/2} and ((d-k)/d)^{(d-k)/2} cannot overcome this exponential gap. Thus I do not believe the central claim of Theorem 1.1 is false; rather, the paper requires a minor but essential correction in Section 4 before it can be accepted as fully correct. The verdict should therefore be CONDITIONAL rather than unconditional ACCEPT, pending verification of the corrected constant inequality. The agreement with the reader is partial because I concur that the external theorem is a risk, but the stronger immediate concern is the internal false bound in Claim 4.2.","tokens_in":36261,"tokens_out":45492,"duration_ms":379090,"concrete_test":"Recompute the key inequality in Claim 4.2 with the exact factor 2^{k+1} instead of 4k: test whether 2^{2k+1}(C0 L'/(d-k))^{(d-k)/2}α ≤ C0^{d/2}(L/d)^{(d-k)/2}α holds for all 1≤k≤d≤2^{-7}L, where L'=L−k log2. If it fails for some admissible range, the proof of Theorem 1.2 requires a substantive change; if it holds, the paper needs only a constant-typo fix.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"In Section 4, proof of Claim 4.2, the triangle inequality argument at (4.5) produces a factor 2^{k+1} (two times the 2^k subsets T⊆S). The text then states 'crudely bounding 2^{k+1} ⩽ 4k' and uses 4k in the subsequent inequality (4.7). This bound is false for k≥3 (for k=3, 16≤12). Since k=|S| can be as large as d, and d can be up to 2^{-7}L, the induction that proves derivative-globalness of W_dΨ_{P,Q}f is invalid as written. The later expansion leading to (4.7) with 8k likewise does not match the exact factor 2^{k+1}·2^k = 2^{2k+1} that would arise if α'=2^kα is substituted. This is an internal inconsistency in the central arithmetic level d inequality (Theorem 1.2), not merely a disagreement with an external source. A quick numerical check with the exact factor suggests the constants C0=2^{13} are still sufficiently large for the required inequality to hold, so the theorem likely survives with a corrected constant; however, the current manuscript needs a revision at exactly the point on which Theorem 1.1 depends.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves Theorem 1.1: if A ⊆ [X] has no two elements differing by a square, then |A| ≤ X exp(-c0 sqrt(log X)), improving the previous Bloom–Maynard bound. The proof follows the density-increment strategy. The main new ingredient is Theorem 1.2, an arithmetic level-d inequality, proved in Section 4 by an inductive argument that imports a global hypercontractivity theorem of Keller, Lifshitz and Marcus. Sections 5–8 develop weight functions, exponential sum estimates, and a multi-stage density increment argument; Section 9 gives a self-contained limitation example for the density-increment method. The paper is written in full detail, with explicit constants and a clearly isolated external input, but one load-bearing inequality in the proof of Claim 4.2 is false as written.","tokens_in":36486,"tokens_out":11154,"duration_ms":103217,"significance":"If the proof is correct, Theorem 1.1 gives the best known quantitative upper bound for the Furstenberg–Sárközy theorem, and Theorem 1.2 is a potentially reusable arithmetic inequality. The paper is unusually careful: constants are explicit, the dependence on the external hypercontractivity result is isolated in Theorem 2.5, and Appendix A addresses the adaptation to complex-valued functions on product groups. Section 9 honestly states limitations of the method. However, the proof as printed contains a false bound at the core of the induction proving Theorem 1.2, so the manuscript is not currently in publishable form; the issue appears local and repairable, not fatal.","major_comments":[{"comment":"The line 'crudely bounding 2^{k+1} ⩽ 4k' is false for k ≥ 3, and k = |S| may be as large as d, which is allowed to be as large as 2^{-7} log(1/α). This bound is load-bearing: it produces the factor 4k used to complete the induction showing that W_d Ψ_{P,Q} f is derivative-global, and thereby Theorem 1.2 and Theorem 1.1. The displayed comparison (4.7) is also inconsistent with the exact factor 2^{k+1}·2^k that would arise after substituting α' = 2^k α. The error is repairable: for k ≥ 1 one has 2^{k+1} ≤ 4^k, and the required comparison becomes 8^k (d/(d-k))^{(d-k)/2} ≤ C_0^{k/2}(L/L')^{(d-k)/2}, which holds since C_0 ≥ 64e. The manuscript should be revised at exactly this point; the central argument appears salvageable with this correction.","section":"Section 4, proof of Claim 4.2, Eqs. (4.5)–(4.7)"}],"minor_comments":[{"comment":"The symbol N should be explicitly the positive integers, or the condition a1 - a2 = n^2 should be paired with n ≥ 1, since n = 0 would make the hypothesis vacuous for any set.","section":"Theorem 1.1 statement"},{"comment":"The displayed identities '‖Tρf‖_m^p = ‖Tρ(f⊗m)‖_p' and '‖f‖_2^m = ‖Tρ(f⊗m)‖_2' appear to contain notational typos; the intended identities are '‖Tρ f‖_p^m = ‖Tρ(f⊗m)‖_p' and '‖f⊗m‖_2 = ‖f‖_2^m'. The tensor-power argument is sound once read this way.","section":"Appendix A"},{"comment":"The proof of Proposition 9.1 invokes a random sampling assertion without proof; since this proposition is a limitation remark rather than part of the main theorem, the omission is acceptable, but a reference or a one-line justification would improve readability.","section":"Section 9"}],"recommendation":"major_revision","confidential_remarks":"The verified issue in Section 4 is a genuine internal error, but it is localized and the constants have enough slack that a corrected bound of the form 2^{k+1} ≤ 4^k should close the gap. I would support acceptance after a careful revision of the induction step in Claim 4.2 and a re-check of the subsequent numerical inequalities. The paper's dependence on Keller–Lifshitz–Marcus is substantial but explicit, and Section 9's limitation discussion is a strength."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The thing to know: this paper proves s(X) ≤ X exp(−c√log X), the best known upper bound for square-difference-free sets, beating Bloom–Maynard and the authors’ own earlier exp(−(log X)^{1/4−o(1)}). The main novelty is the arithmetic level d inequality (Theorem 1.2), proved via global hypercontractivity from Keller–Lifshitz–Marcus. That is a genuinely new ingredient in additive number theory, and the paper does a good job of exposing it.\n\nWhat it does well: the proof is written in full detail, with explicit constants, a transparent record of earlier versions, and an honest section on limitations of the density increment method. The exponential sum estimates and the random sparsification analysis in Sections 5–8 look careful and sound. The paper deserves credit for being self-contained modulo the stated external theorem.\n\nThe soft spot is real. In the proof of Claim 4.2, the triangle inequality at (4.5) produces a factor 2^{k+1}; the text then says “crudely bounding 2^{k+1} ≤ 4k”, which is false for k ≥ 3. Since k = |S| can be as large as d, and d can be as large as 2^{-7} log(1/α), k = 3 is well within range. This step feeds the induction proving Theorem 1.2, so the manuscript as written is not fully correct. The stress-test note’s diagnosis holds up.\n\nThe good news: this looks like a constants-level oversight, not a structural flaw. Replacing 4k by the honest 2^{k+1}, or absorbing the extra 2^k into a slightly larger C0, should make the induction go through. I did not re-verify every later constant, but the fix is plausible and the overall strategy survives. The reliance on [13, Corollary 4.7] for complex-valued functions on products of distinct cyclic groups is heavy, but Appendix A sketches the adaptation responsibly.\n\nWho this is for: additive number theorists, and anyone using density increments or global hypercontractivity in diophantine problems. It deserves a serious referee; the referee should request a corrected proof of Claim 4.2 before acceptance. My own view: the main theorem is almost certainly true, but the current text needs that revision.","headline":"Strong new bound on Furstenberg–Sárközy with a repairable gap in the key inductive claim — worth serious referee attention, but the current text needs a fix at (4.5)–(4.7).","tokens_in":37097,"tokens_out":2217,"would_cite":true,"duration_ms":22164,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11B30","11L07","11N37"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that every square-difference-free subset of {1,...,X} has size at most X exp(-c0 sqrt(log X)), the strongest known quantitative form of the square-difference theorem.","keywords":["Furstenberg–Sárközy theorem","square differences","density increment","global hypercontractivity","Fourier level inequality","additive combinatorics","intersective sets"],"falsifier":"Verify the imported hypercontractivity inequality for a concrete complex-valued $(r,\\gamma)$-derivative-global function on a product of distinct cyclic groups $\\mathbb{Z}/q\\mathbb{Z}$, with $m=\\lceil r^{-2}\\rceil$, $p=2m$, and $\\rho=1/(20\\sqrt{m})$; a single violation at these parameters invalidates Claim 4.2 and with it Theorem 1.2. A more direct but less structured check is to search for square-difference-free sets of size exceeding $X\\exp(-c_0\\sqrt{\\log X})$, which the theorem asserts do not exist.","tokens_in":36021,"feed_emoji":"🔢","tokens_out":9925,"duration_ms":85621,"temperature":0.7,"pith_summary":"The paper proves a new upper bound for the square-difference problem: if $A\\subseteq\\{1,\\dots,X\\}$ contains no two elements differing by a nonzero square, then $|A|\\le X\\exp(-c_0\\sqrt{\\log X})$ for an absolute constant $c_0>0$. This replaces previous bounds that saved only iterated-logarithm factors, and it is the first bound of this strength with exponent $1/2$ in place of smaller exponents. The proof uses the density increment strategy, powered by a new arithmetic level-$d$ inequality that controls sums of squared Fourier coefficients at rationals with many small prime factors. The decisive input is a global hypercontractivity theorem for functions on products of cyclic groups, imported from another work and adapted to complex-valued functions in an appendix. A final section shows that the density increment method as formulated cannot yield a bound better than $X\\exp(-c(\\log X)^{3/4})$.","feed_headline":"No-square-difference sets capped at X exp(-c sqrt log X)","feed_subtitle":"A Fourier-level inequality plus a noise-smoothed norm bound beats all previous records.","key_machinery":"The load-bearing object is the Fourier level-$d$ operator $W_d$, which keeps only frequencies whose support has exactly $d$ coordinates on the product group $G_Q=\\prod_{q\\in Q}\\mathbb{Z}/q\\mathbb{Z}$. The proof shows that, after lifting a function from a long arithmetic progression to $G_Q$, an 'integer-global' function — one with no dense subprogression cut out by few moduli — becomes a derivative-global function, meaning every iterated Laplacian derivative has controlled $L^2$ norm. The identity $D_{S,x}W_d=W_{d-|S|}D_{S,x}$ allows induction on $d$, and a global hypercontractivity theorem (Theorem 2.5) is applied with $p=2m$, $m=\\lceil \\log(1/\\alpha)/d\\rceil$, and noise rate $\\rho=1/(20\\sqrt{m})$. A lifting lemma transfers the estimate back from $G_Q$ to the original progression with an error term $\\alpha^{2m}X^{-1/4}$.","core_discovery":"The paper's main discovery is Theorem 1.1: there is a constant $c_0>0$ such that for all $X\\ge 10$, every square-difference-free $A\\subseteq[X]$ satisfies $|A|\\le X\\exp(-c_0\\sqrt{\\log X})$. The engine behind it is Theorem 1.2, an arithmetic level-$d$ inequality. For a bounded function $f$ on $[X]$, pairwise coprime moduli $q\\in Q$, and $d\\le 2^{-7}\\log(1/\\alpha)$, either the sum of $|\\hat f(a/\\prod_{q\\in S}q)|^2$ over all $S\\subseteq Q$ with $|S|=d$ is at most $(C_0\\log(1/\\alpha)/d)^d\\alpha^2X^2$, or some subprogression with common difference a product of at most $2\\log(1/\\alpha)$ elements of $Q$ has density at least $2^{|S|}\\alpha$. Feeding this dichotomy into a carefully staged density increment argument yields the square-root-of-log bound.","pith_inferences":["The same level-$d$ machinery should transfer to differences $P(n)$ for other intersective polynomials, with only the exponential-sum estimates changing; the paper says this is likely but does not carry it out.","A quantitative strengthening of the imported hypercontractivity theorem for products of unequal cyclic groups would likely improve the $1/2$ exponent in Theorem 1.1; the proof's bottleneck sits exactly there.","The Section 9 limitation suggests that any future bound stronger than $X\\exp(-c(\\log X)^{3/4})$ must abandon the density-increment template and exploit structure beyond Fourier correlations with subprogressions."],"forward_implications":["For every $X\\ge 10$, the largest square-difference-free subset of $\\{1,\\dots,X\\}$ has size at most $X\\exp(-c_0\\sqrt{\\log X})$.","The arithmetic level-$d$ inequality gives a quantitative dichotomy: a set either has small Fourier energy at rationals with many prime factors, or it has a long square-common-difference subprogression of density $2^{|S|}\\alpha$.","The proof introduces a factorization of moduli into cube-free and cube-full parts plus a random sparsification step; these devices are what allow the level-$d$ inequality to be applied efficiently across different prime-factorization types.","Within the density-increment framework used here and in earlier work, no bound better than $X\\exp(-c(\\log X)^{3/4})$ can be obtained; the paper's Section 9 example shows the limit of the method itself."],"supporting_citations":[{"why":"Supplies the global hypercontractivity inequality used as the black-box engine for the arithmetic level-d bound.","marker":"[13]"},{"why":"Gives the previous best upper bound, the benchmark this paper improves.","marker":"[3]"},{"why":"Introduced the density-increment Fourier framework and an earlier iterated-logarithm estimate.","marker":"[16]"},{"why":"Supplies the original Fourier-analytic proof of the qualitative theorem and the first explicit rate.","marker":"[19]"},{"why":"Provides the earlier hypercontractivity machinery that could give a weaker quasi-polynomial bound in place of the main input.","marker":"[12]"},{"why":"Established the qualitative form of the theorem by ergodic methods, which the quantitative bound extends.","marker":"[6]"},{"why":"Gives the lower-bound construction showing the answer is at least $X^{1/2+0.233}$, the main constraint an upper bound must respect.","marker":"[18]"}],"fun_headline_variants":["Square-difference-free sets: new bound exp(-c sqrt(log N))","Furstenberg-Sárközy theorem: density improved to exp(-c sqrt(log N))","New density bound for square-difference-free sets: exp(-c sqrt(log N))","Best known bound for Furstenberg-Sárközy: N exp(-c sqrt(log N))","Square-difference-free sets capped at N exp(-c sqrt(log N))"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is an imported theorem about how noise shrinks the high-frequency parts of functions on products of cyclic groups; the paper adapts it to complex-valued functions and specific parameters, but it does not prove that theorem. If the imported inequality is false in the needed parameter range, the arithmetic level-$d$ inequality and the main bound collapse.","fun_headline_variants_meta":{"raw":{"variants":["Square-difference-free sets: new bound exp(-c sqrt(log N))","Furstenberg-Sárközy theorem: density improved to exp(-c sqrt(log N))","New density bound for square-difference-free sets: exp(-c sqrt(log N))","Best known bound for Furstenberg-Sárközy: N exp(-c sqrt(log N))","Square-difference-free sets capped at N exp(-c sqrt(log N))"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00151,"raw_usage":{"total_tokens":5974,"prompt_tokens":785,"completion_tokens":5189,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":401,"completion_tokens_details":{"reasoning_tokens":5076}},"tokens_in":401,"tokens_out":5189,"duration_ms":38808,"temperature":1.0,"reasoning_tokens":5076,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T12:05:33.734517+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Verify the imported hypercontractivity inequality for a concrete complex-valued $(r,\\gamma)$-derivative-global function on a product of distinct cyclic groups $\\mathbb{Z}/q\\mathbb{Z}$, with $m=\\lceil r^{-2}\\rceil$, $p=2m$, and $\\rho=1/(20\\sqrt{m})$; a single violation at these parameters invalidates Claim 4.2 and with it Theorem 1.2. A more direct but less structured check is to search for square-difference-free sets of size exceeding $X\\exp(-c_0\\sqrt{\\log X})$, which the theorem asserts do not exist.","supporting_citations":[{"cited_title":"Bloom and James Maynard, A new upper bound for sets with no square differences , Compos","cited_arxiv_id":null,"evidence_quote":"Gives the previous best upper bound, the benchmark this paper improves."},{"cited_title":"London Math","cited_arxiv_id":null,"evidence_quote":"Introduced the density-increment Fourier framework and an earlier iterated-logarithm estimate."},{"cited_title":"I , Acta Math","cited_arxiv_id":null,"evidence_quote":"Supplies the original Fourier-analytic proof of the qualitative theorem and the first explicit rate."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the earlier hypercontractivity machinery that could give a weaker quasi-polynomial bound in place of the main input."},{"cited_title":"Analyse Math","cited_arxiv_id":null,"evidence_quote":"Established the qualitative form of the theorem by ergodic methods, which the quantitative bound extends."},{"cited_title":"Ruzsa, Difference sets without squares , Period","cited_arxiv_id":null,"evidence_quote":"Gives the lower-bound construction showing the answer is at least $X^{1/2+0.233}$, the main constraint an upper bound must respect."}],"review_version":1}