An explicit version of Bombieri's log-free density estimate and Sárközy's theorem for shifted primes

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 $N\ge 2$ is an integer, $A\subseteq \{1,\dots ,N\}$, 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 Sárközy.

24 pages. v4: Incorporates referee comments

References in corpus (3)

• An explicit bound for the least prime ideal in the Chebotarev density theorem
• On a theorem of Sárközy for difference sets and shifted primes
• Explicit bounds on exceptional zeroes of Dirichlet $L$-functions

Cited by in corpus (1)

• Explicit Deuring-Heilbronn phenomenon for Dirichlet $L$-functions