Visiting early at prime times

TL;DR

This paper proves the existence of arithmetic progressions with prime numbers where the m-th least prime is bounded by a constant times the common difference, using a Maynard--Tao sieve variant, and extends the results to dynamical systems.

Contribution

It introduces a new application of the Maynard--Tao sieve to find prime progressions with controlled prime positions and generalizes the results to dynamical systems.

Findings

Existence of prime progressions with the m-th prime bounded by a constant times the common difference.

Extension of prime progression results to dynamical systems.

Dependence of results on first return times in dynamical systems.

Abstract

Given an integer $m \geq 2$ and a sufficiently large $q$, we apply a variant of the Maynard--Tao sieve weight to establish the existence of an arithmetic progression with common difference $q$ for which the $m$-th least prime in such progression is $\ll_m q$, which is best possible. As we vary over progressions instead of fixing a particular one, the nature of our result differs from others in the literature. 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.