  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.
