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New bounds for the Furstenberg-S\'ark\"ozy theorem

T0 review · 1 major / 3 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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). read the letter →

arxiv 2411.17448 v2 pith:REAQ32WJ submitted 2024-11-26 math.NT

Ben Green , Mehtaab Sawhney This is my paper · ORCID
classification math.NT MSC 11B3011L0711N37
keywords Furstenberg–SárközytheoremsquaredifferencesdensityincrementglobalhypercontractivityFourierlevelinequalityadditivecombinatoricsintersectivesets
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 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 $X\exp(-c(\log X)^{3/4})$.

What carries the argument

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 $\alpha^{2m}X^{-1/4}$.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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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

1 major / 3 minor

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.

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 (1)
  1. [Section 4, proof of Claim 4.2, Eqs. (4.5)–(4.7)] 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.
minor comments (3)
  1. [Theorem 1.1 statement] 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.
  2. [Appendix A] 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.
  3. [Section 9] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main result is self-contained modulo an external hypercontractivity theorem imported from Keller–Lifshitz–Marcus.

full rationale

The derivation chain is not circular. Theorem 1.1 is deduced from Proposition 6.1, which in turn rests on the arithmetic level d inequality Theorem 1.2. The proof of Theorem 1.2 is carried out in Section 4 through an induction (Claim 4.2), and its only imported input is Theorem 2.5, stated explicitly as a version of Corollary 4.7 of Keller–Lifshitz–Marcus [13]. That theorem is external to the present paper and does not involve the target result or the authors’ own fitted constants. Appendix A adapts it to complex-valued functions on product groups via tensor-power tricks, but the underlying hypercontractivity bound remains independent evidence rather than a self-citation. The paper’s constants C0, c, C, W, C1 are chosen to satisfy explicit inequalities in the course of the proof (for example, C0 “much bigger than 64e” in Claim 4.2), not fitted to reproduce the final bound. The only self-citation, reference [7], is explicitly marked “not for publication” and is used for historical context and for a detailed proof of an intermediate exponent from an earlier version; it is not a load-bearing premise for Theorem 1.1. The reviewer-flagged inequality “crudely bounding 2^{k+1} ≤ 4k” at equation (4.7) is a local mathematical correctness concern about the manuscript as written, not a circularity: it does not make the target result equivalent to an input or fitted parameter. Therefore the circularity score is 0.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The central claim rests on one heavy external theorem (KLM hypercontractivity) and otherwise standard results in analytic number theory. No data are fitted, no invented entities (particles, forces, etc.) are introduced.

assumptions (2)
  • domain assumption Global hypercontractivity for product spaces (Theorem 2.5, from Keller-Lifshitz-Marcus [13, Corollary 4.7])
    The proof of Theorem 1.2 (via Claim 4.2) applies this theorem to the lifted function W_d(f~) to obtain the key bound (4.15) and the contradiction. The authors adapt it to complex-valued functions on non-identical factors in Appendix A.
  • standard math Standard analytic number theory tools (Weyl's inequality, Dirichlet approximation, Gauss sum estimates, prime number theorem)
    Used throughout Sections 5 and 7, e.g., Lemma 5.2 (Weyl), Lemma 5.5 (Gauss sums), and the lower bound for the product of the first k primes in Section 7.1.

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Pith. "Pith review of New bounds for the Furstenberg-S\'ark\"ozy theorem." pith.science (2026). https://pith.science/paper/REAQ32WJ

@misc{pith2026241117448,
  author       = {Pith},
  title        = {Pith review of: New bounds for the Furstenberg-S\'ark\"ozy theorem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/REAQ32WJ}},
  note         = {Machine review of arXiv:2411.17448}
}
abstract

Suppose that $A \subset \{1,\dots, N\}$ has no two elements differing by a square. Then $|A| \ll N e^{-c\sqrt{\log N}}$.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. The sharp exponent for the minimal distance problem

    math.CO 2026-07 accept novelty 8.0 of 10

    The minimal distance problem has sharp exponent 2/3: lower-bound configurations of size n with separation n^{-2/3−o(1)} are constructed via trace-zero lattices in high-degree totally real number fields.

Reference graph

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