Dense clusters of primes in subsets

Abstract

We prove a generalization of the author’s work to show that any subset of the primes which is ‘well distributed’ in arithmetic progressions contains many primes which are close together. Moreover, our bounds hold with some uniformity in the parameters. As applications, we show there are infinitely many intervals of length$(\log x)^{{\it\epsilon}}$containing$\gg _{{\it\epsilon}}\log \log x$primes, and show lower bounds of the correct order of magnitude for the number of strings of$m$congruent primes with$p_{n+m}-p_{n}\leqslant {\it\epsilon}\log x$.