Waring's number for large subgroups of $\mathbb Z_{p}^{*}$
Volume 163 / 2014
Acta Arithmetica 163 (2014), 309-325
MSC: Primary 11L07; Secondary 11B30, 11P05.
DOI: 10.4064/aa163-4-2
Abstract
Let $p$ be a prime, $\mathbb Z_p$ be the finite field in $p$ elements, $k$ be a positive integer, and $A$ be the multiplicative subgroup of nonzero $k$th powers in $\mathbb Z_p$. The goal of this paper is to determine, for a given positive integer $s$, a value $t_s$ such that if $|A|\gg t_s$ then every element of $\mathbb Z_p$ is a sum of $s$ $k$th powers. We obtain $t_4 = p^{{22/39}+\epsilon }$, $t_5= p^{{15/29}+\epsilon }$ and for $s \ge 6$, $t_s= p^{\frac {9s+45}{29s+33}+\epsilon }$. For $s \ge 24$ further improvements are made, such as $t_{32}=p^{{5/16} + \epsilon }$ and $t_{128} = p^{{1/4}}$.
