On short products of primes in arithmetic progressions

We give several families of reasonably small integers $k,\mathrm{\ell }\ge 1$ and real positive $\alpha ,\beta \le 1$, such that the products $p_1\dots p_{k}s$, where $p_1,\dots ,p_k\le m^{\alpha }$ are primes and $s\le m^{\beta }$ is a product of at most $\mathrm{\ell }$ primes, represent all reduced residue classes modulo $m$. This is a relaxed version of the still open question of P. Erdos, A. M. Odlyzko and A. Sarkozy (1987), that corresponds to $k=\mathrm{\ell }=1$ (that is, to products of two primes). In particular, we improve recent results of A. Walker (2016).