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

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 $\mathrm{\mid }A\mathrm{\mid }\ll N^{1-1\mathrm{/}{10}^{18}}$. This strengthens a theorem of Sárközy.