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Aled Walker, A multiplicative analogue of Schnirelmann's theorem, Bulletin of the London Mathematical Society, Volume 48, Issue 6, December 2016, Pages 1018–1028, https://doi.org/10.1112/blms/bdw062
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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 |$(\mathbb {Z}/ q\mathbb {Z})^{\times }$| 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$|, viewed naturally as a subset of |$(\mathbb {Z}/ q\mathbb {Z})^{\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^{-{1}/{4}+\epsilon },$| there exists a constant |$k$| depending only on |$\epsilon $| such that |$P_\eta ^{(k)}=(\mathbb {Z}/ q\mathbb {Z})^{\times }$|. Erdös conjectured that, for |$\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.
