Subproducts of small residue classes

Abstract

For any prime $p$, let $y(p)$ denote the smallest integer $y$ such that every reduced residue class $(\mathrm{mod}p)$ is represented by the product of some subset of $\{1,\dots ,y\}$. It is easy to see that $y(p)$ is at least as large as the smallest quadratic nonresidue $(\mathrm{mod}p)$; we prove that $y(p){\ll }_{\epsilon }{p}^{1/(4\sqrt{e})+\epsilon }$, thus strengthening Burgess's classical result. This result is of intermediate strength between two other results, namely Burthe's proof that the multiplicative group $(\mathrm{mod}p)$ is generated by the integers up to $O_{\epsilon }(p^{1/(4\sqrt{e})+\epsilon }$, and Munsch and Shparlinski's result that every reduced residue class $(\mathrm{mod}p)$ is represented by the product of some subset of the primes up to $O_{\epsilon }(p^{1/(4\sqrt{e})+\epsilon }$. Unlike the latter result, our proof is elementary and similar in structure to Burgess's proof for the least quadratic nonresidue.

7 pages