Visiting early at prime times

Abstract

Given an integer $m⩾2$ and a sufficiently large $q$, we prove that there always exists an arithmetic progression with common difference $q$ for which the $m$-th least prime in such progression is $\ll q\exp (3.8075m)$. This is obtained using a minorant for the indicator of primes in the Maynard--Tao sieve, in parallel with Baker--Irving and Stadlmann's work on bounded gaps between primes. The dependence on $q$ is best possible. Furthermore, we generalize our result to dynamical systems. The quality of the result depends crucially on the first return time, which we illustrate in the case of Diophantine approximation.

arXiv comment18 pages. Substantial revision improving the dependence on $m$