S\'ark\"ozy's theorem in $\mathbf{F}_2[x]$

TL;DR

This paper proves an unconditional analogue of Green's theorem in the polynomial ring over $ extbf{F}_2$, showing that large subsets necessarily contain differences of the form $p-1$ for some prime polynomial, with an improved exponent.

Contribution

It establishes an unconditional version of Sárközy's theorem in $ extbf{F}_2[x]$, improving the exponent from previous conditional results.

Findings

Unconditional proof of Sárközy's theorem in $ extbf{F}_2[x]$

Improved exponent from 11/12 + ε to 7/8 + ε

Large subsets contain polynomial differences of the form p-1

Abstract

Green showed that, conditional on GRH, a subset $A \subseteq [N]$ with $\mid A \mid \gg_{\epsilon} N^{\frac{11}{12}+\epsilon}$ 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 $\frac{7}{8}+\epsilon$.