  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.
