Dense clusters of primes in subsets

TL;DR

This paper demonstrates that well-distributed prime subsets contain dense clusters of primes, establishing new bounds and applications for small intervals and strings of congruent primes.

Contribution

It generalizes previous work to show dense prime clusters in well-distributed subsets, with uniform bounds and new applications for small prime intervals and strings.

Findings

Infinitely many intervals of length $( ext{log } x)^{ ext{epsilon}}$ contain many primes.

Lower bounds for the number of strings of congruent primes are established.

Results hold uniformly across parameters.

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 $(\log{x})^{\epsilon}$ containing $\gg_\epsilon \log\log{x}$ primes, and show lower bounds of the correct order of magnitude for the number of strings of $m$ congruent primes with $p_{n+m}-p_n\le \epsilon\log{x}$.