A multiplicative analogue of Schnirelmann's theorem

Abstract: The classical theorem of Schnirelmann states that the primes are an additive basis for the integers. In this paper we consider the analogous multiplicative setting of the cyclic group $\left(\mathbb{Z}/ q\mathbb{Z}\right)^{\times}$, and prove a similar result. For all suitably large primes $q$ we define $P_\eta$ to be the set of primes less than $\eta q$, viewed naturally as a subset of $\left(\mathbb{Z}/ q\mathbb{Z}\right)^{\times}$. Considering the $k$-fold product set $P_\eta^{(k)}={p_1p_2\cdots p_k:p_i\in P_\eta }$, we show that for $\eta \gg q^{-\frac{1}{4}+\epsilon}$ there exists a constant $k$ depending only on $\epsilon$ such that $P_\eta^{(k)}=\left(\mathbb{Z}/ q\mathbb{Z}\right)^{\times}$. Erd\H{o}s conjectured that for $\eta = 1$ the value $k=2$ should suffice: although we have not been able to prove this conjecture, we do establish that $P_1 ^{(2)}$ has density at least $\frac{1}{64}(1+o(1))$. We also formulate a similar theorem in almost-primes, improving on existing results.