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We make explicit Bombieri's refinement of Gallagher's log-free "large sievedensity estimate near $\sigma = 1$" for Dirichlet $L$-functions. We use thisestimate 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 ofS\'ark\"ozy.

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