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 σ = 1 {\sigma=1} for Dirichlet L-functions. We use this estimate and recent work of Green to prove that if N ≥ 2 {N\geq 2} is an integer, A S { 1, ..., N } {A\subseteq\{1,\ldots,N\}}, and for all primes p no two elements in A differ by p - 1 {p-1}, then | A | N 1 - 10 - 18 {|A|\ll N^{1-10^{-18}}}. This strengthens a theorem of Sárközy.