The distribution of smooth numbers in arithmetic progressions

For a wide range of $x$ and $y$ we show that ${\Cal S}(x,y)$, the set of integers below $x$ composed only of prime factors below $y$, is equidistributed in the reduced residue classes $\pmod q$ for all $q<y^{4\sqrt{e}-\epsilon}$. This improves earlier work of Granville; any improvement of this range of $q$ would have interesting consequences for Vinogradov's conjecture on the least quadratic non-residue. For larger ranges of $q$ we prove the existence of a large subgroup of the group of reduced residues such that ${\Cal S}(x,y)$ is equidistributed within cosets of that subgroup.