When the sieve works

TL;DR

This paper investigates conditions under which sieving integers with certain prime sets yields an expected count of remaining numbers, extending results to include larger primes using additive combinatorics.

Contribution

It provides the first general results for sieving intervals with primes including those larger than rac{1}{2}rac{rac{1}{2}rac{x}{rac{1}{2}rac{x}{rac{1}{2}rac{x}} including some in rac{rac{rac{1}{2}rac{x}{rac{1}{2}rac{x}{rac{1}{2}rac{x}} using methods inspired by additive combinatorics.

Findings

Established conditions for prime sets to produce expected sieving results

Extended sieving results to include primes in rac{rac{rac{1}{2}rac{x}{rac{1}{2}rac{x}{rac{1}{2}rac{x}}

Applied additive combinatorics techniques to sieve analysis

Abstract

We are interested in classifying those sets of primes $\mathcal{P}$ such that when we sieve out the integers up to $x$ by the primes in $\mathcal{P}^c$ we are left with roughly the expected number of unsieved integers. In particular, we obtain the first general results for sieving an interval of length $x$ with primes including some in $(\sqrt{x},x]$, using methods motivated by additive combinatorics.