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.
Categories: math.NT math.CO