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

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 $\frac{7}{8}+\epsilon$.

15 pages