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.