Bounded Gaps Between Products of Distinct Primes

TL;DR

This paper extends the Maynard-Tao sieve method to establish the smallest known bounded gaps between numbers that are products of a fixed number of distinct primes, applicable to certain prime subsets.

Contribution

It introduces an adaptation of the Maynard-Tao sieve to find bounded gaps between products of r distinct primes, improving previous results.

Findings

Established asymptotically optimal bounds for gaps between such products.

Applicable to positive-density prime subsets with specific distribution properties.

Improved upon previous bounds by Thorne and Sono.

Abstract

Let $r \ge 2$ be an integer. We adapt the Maynard-Tao sieve to produce the asymptotically best-known bounded gaps between products of $r$ distinct primes. Our result applies to positive-density subsets of the primes that satisfy certain equidistribution conditions. This improves on the work of Thorne and Sono.