Products of primes in arithmetic progressions

A conjecture of Erdős states that for every large enough prime q, every reduced residue class modulo q is the product of two primes less than q. I will discuss my on-going work with Kaisa Matomäki establishing among other things a ternary variant of Erdős' conjecture, namely that every reduced residue class modulo q is the product of three primes less than q. The proof is based on a multiplicative transference principle, Kneser's theorem, and bounds for the least primes in cosets of small index subgroups.