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New bounds for the Furstenberg-Sárközy theorem

Ben Green, Mehtaab Sawhney

TL;DR

The paper proves a new explicit bound for the Furstenberg–Sárközy problem: there exists $c_0>0$ such that for all $X\ge 10$, any $A\subset [X]$ with no two elements differing by a square satisfies $|A| \le X e^{-c_0 \sqrt{\log X}}$. The authors develop a density-increment framework anchored by a novel arithmetic level-$d$ inequality derived from a global hypercontractivity result of Keller–Lifshitz–Marcus (KLM23) via a lifting construction to product groups; they control square-difference structures using smooth weights on squares and Weyl-type exponential-sum bounds, and translate level-d information into arithmetic-energy bounds $E_{t,k,\vec{\beta}}$, eventually iterating density increments to reach the main bound. A key technical engine is the lifting of functions from $[X]$ to a finite abelian product group $G_{\mathcal{Q}}$ so that hypercontractive estimates can be applied through derivative-global analysis. The work also discusses intrinsic limitations of the density-increment approach, constructing a barrier to strictly faster than quasi-polynomial improvements within this framework. Overall, the results sharpen the quantitative understanding of Furstenberg–Sárközy-type problems and provide a flexible method potentially adaptable to broader polynomial difference settings.

Abstract

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

New bounds for the Furstenberg-Sárközy theorem

TL;DR

The paper proves a new explicit bound for the Furstenberg–Sárközy problem: there exists such that for all , any with no two elements differing by a square satisfies . The authors develop a density-increment framework anchored by a novel arithmetic level- inequality derived from a global hypercontractivity result of Keller–Lifshitz–Marcus (KLM23) via a lifting construction to product groups; they control square-difference structures using smooth weights on squares and Weyl-type exponential-sum bounds, and translate level-d information into arithmetic-energy bounds , eventually iterating density increments to reach the main bound. A key technical engine is the lifting of functions from to a finite abelian product group so that hypercontractive estimates can be applied through derivative-global analysis. The work also discusses intrinsic limitations of the density-increment approach, constructing a barrier to strictly faster than quasi-polynomial improvements within this framework. Overall, the results sharpen the quantitative understanding of Furstenberg–Sárközy-type problems and provide a flexible method potentially adaptable to broader polynomial difference settings.

Abstract

Suppose that has no two elements differing by a square. Then .

Paper Structure

This paper contains 23 sections, 19 theorems, 205 equations.

Table of Contents

  1. Introduction
  2. History
  3. Arithmetic level $d$ inequality
  4. Structure of the paper
  5. Notation
  6. Acknowledgements
  7. Statement of hypercontractivity results
  8. Liftings
  9. Proof of the arithmetic level inequality
  10. Weight functions and exponential sums
  11. A smooth weight on the squares
  12. Density increment arguments
  13. Fourier coefficients and density increments
  14. First steps towards the density increment
  15. Applying the level inequality: bounds for arithmetic energies
  16. ...and 8 more sections

Key Result

Theorem 1.1

There exists $c_0 >0$ for which the following holds. Let $X \geqslant 10$ and suppose that $A\subseteq [X]$ has no solutions to $a_1-a_2 = n^2$ with $a_1,a_2\in A$ and $n\in \mathbf{N}$.

Theorems & Definitions (46)

  • Theorem 1.1
  • Theorem 1.2
  • Remark
  • Lemma 2.1
  • proof
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Theorem 2.5
  • ...and 36 more