Few small primes suffice to cover unit groups

Klurman, Shparlinski and Teräväinen showed that there is a set of at most $(\log Q{)}^{1+\epsilon }$ primes of polylogarithmic size whose subset products cover the unit group $(\mathbb{Z}\mathrm{/}q\mathbb{Z}{)}^{\times }$ for almost all moduli $q\le Q$. We show that $(1\mathrm{/}\log 2+o(1))\log Q$ primes suffice, which is optimal to first order, since $m$ primes have at most $2^m$ subset products. The proof combines a doubling argument of Erdős–Rényi type for random subset products in a finite abelian group, modified to tolerate a small set of exceptional characters, with a one-time ``repair’’ of the few primitive characters whose $L$-functions have zeros near $s=1$. An exact weighted repair gives a variant with at most $(2\mathrm{/}\log 2+o(1))\log Q$ primes and an exceptional set described explicitly.