Bombieri-Vinogradov type theorems for sparse sets of moduli

We prove two estimates for the Barban--Davenport--Halberstam type variance of a general complex sequence in arithmetic progressions. The proofs are elementary, and our estimates are capable of yielding an asymptotic for the variance when the sequence is sufficiently nice, and is either somewhat sparse or is sufficiently like the integers in its divisibility by small moduli. As a concrete application, we deduce a Barban--Davenport--Halberstam type variance asymptotic for the $y$-smooth numbers less than $x$, on a wide range of the parameters. This addresses a question considered by Granville and Vaughan.