Weighted sieves with switching

TL;DR

This paper explores the use of weighted sieves with switching principles to detect numbers with few prime factors, improving results in Diophantine approximation involving primes and almost primes.

Contribution

It introduces how different sieve weights function in symmetric two-variable problems, enhancing detection of primes paired with almost primes, and improves existing bounds in related problems.

Findings

Improved bounds for Diophantine approximation with primes and almost primes.

Enhanced understanding of sieve weights in symmetric two-variable problems.

Demonstrated level of distribution at least 0.267 for certain prime pairs.

Abstract

Weighted sieves are used to detect numbers with at most $S$ prime factors with $S \in \mathbb{N}$ as small as possible. When one studies problems with two variables in somewhat symmetric roles (such as Chen primes, that is primes $p$ such that $p+2$ has at most two prime factors), one can utilize the switching principle. Here we discuss how different sieve weights work in such a situation, concentrating in particular on detecting a prime along with a product of at most three primes. As applications, we improve on the works of Yang and Harman concerning Diophantine approximation with a prime and an almost prime, and prove that, in general, one can find a pair $(p, P_3)$ when both the original and the switched problem have level of distribution at least $0.267$.