An explicit version of Bombieri's log-free density estimate and S\'ark\"ozy's theorem for shifted primes

TL;DR

This paper provides an explicit version of Bombieri's density estimate for Dirichlet L-functions and applies it, along with Green's recent work, to improve bounds in a Sárközy-type theorem concerning shifted primes.

Contribution

It makes Bombieri's refinement explicit and uses it to strengthen Sárközy's theorem on differences in sets avoiding shifted primes.

Findings

Established an explicit form of Bombieri's density estimate.

Proved a new upper bound on the size of sets avoiding differences of the form p-1.

Extended Sárközy's theorem with improved quantitative bounds.

Abstract

We make explicit Bombieri's refinement of Gallagher's log-free "large sieve density estimate near $\sigma = 1$" for Dirichlet $L$-functions. We use this estimate and recent work of Green to prove that if $N\geq 2$ is an integer, $A\subseteq\{1,\ldots,N\}$, and for all primes $p$ no two elements in $A$ differ by $p-1$, then $|A|\ll N^{1-1/10^{18}}$. This strengthens a theorem of S\'ark\"ozy.