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Sárközy's theorem in $\mathbf{F}_2[x]$

Aleksandra Kowalska

TL;DR

This work establishes an unconditional Sárközy-type bound in the function field setting over $\mathbf{F}_2[x]$: if a subset $A$ of polynomials with degree bound $N$ has $A-A$ avoiding any $r-1$ for irreducible $r$, then $|A|$ is at most $2^{(7/8+\varepsilon)N}$ up to constants. The authors adapt a Fourier-analytic, van der Corput framework to function fields, leveraging a two-term sieve via $\Psi$ and $\Psi'$ and a streamlined major/minor arc analysis that obviates smoothing. The argument hinges on carefully controlling Fourier coefficients, truncations, and the behavior on major arcs, culminating in unconditional control analogous to the integer case but with a sharper exponent thanks to the binary function-field structure. These techniques also connect to broader results for general finite fields and highlight a clean, largely self-contained two-sieve approach in the function-field setting.

Abstract

Green showed that, conditional on GRH, a subset $A \subseteq [N]$ with $\mid A \mid \gg_ε N^{\frac{11}{12}+ε}$ must contain two elements whose difference is $p-1$ for $p$ a prime. We prove an analogous unconditional result for $\mathbf{F}_2[x]$, improving the exponent to $\frac{7}{8}+ε$.

Sárközy's theorem in $\mathbf{F}_2[x]$

TL;DR

This work establishes an unconditional Sárközy-type bound in the function field setting over : if a subset of polynomials with degree bound has avoiding any for irreducible , then is at most up to constants. The authors adapt a Fourier-analytic, van der Corput framework to function fields, leveraging a two-term sieve via and and a streamlined major/minor arc analysis that obviates smoothing. The argument hinges on carefully controlling Fourier coefficients, truncations, and the behavior on major arcs, culminating in unconditional control analogous to the integer case but with a sharper exponent thanks to the binary function-field structure. These techniques also connect to broader results for general finite fields and highlight a clean, largely self-contained two-sieve approach in the function-field setting.

Abstract

Green showed that, conditional on GRH, a subset with must contain two elements whose difference is for a prime. We prove an analogous unconditional result for , improving the exponent to .

Paper Structure

This paper contains 11 sections, 18 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. Fourier analysis in function fields - background
  3. Van der Corput properties
  4. Definitions and outline of the proof of Theorem \ref{['original_2_1']}
  5. Bounds on Fourier coefficients
  6. Proof of Proposition \ref{['original_3_1']}
  7. Proof of Proposition \ref{['original_3_2']}
  8. Proof of Proposition \ref{['original_3_3']}
  9. Proof of Proposition \ref{['original_3_4']}
  10. Proof of Proposition \ref{['original_3_6']}
  11. Some properties of function fields, analogous to the integers

Key Result

Theorem 1.1

Let $A \subset \{f \in \mathbf{F}_2[x]: \deg(f) < N\}$ be such that $A-A$ contains no polynomial $r-1$, for $r$ irreducible. Then $|A| \ll_{\varepsilon} 2^{(\frac{7}{8}+\varepsilon)N}$.

Theorems & Definitions (26)

  • Theorem 1.1
  • Remark
  • Definition
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Theorem 3.1
  • proof : Proof that Theorem \ref{['original_2_1']} implies Theorem \ref{['original_1_1']}
  • Remark
  • Proposition 4.1
  • ...and 16 more