WHEN THE SIEVE WORKS

Abstract

<p> We are interested in classifying those sets of primes P such that when we sieve out the integers up to x by the primes in 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 (root x, x], using methods motivated by additive combinatorics.</p>