A note on sumsets of subgroups in ${\mathbb Z}_{p}^{*}$
Volume 161 / 2013
Acta Arithmetica 161 (2013), 387-395
MSC: Primary 11B30; Secondary 11B13.
DOI: 10.4064/aa161-4-5
Abstract
Let $A$ be a multiplicative subgroup of $\mathbb Z_p^*$. Define the $k$-fold sumset of $A$ to be $kA=\{x_1+\dots +x_k:x_i \in A$, $1\leq i\leq k\}$. We show that $6A\supseteq \mathbb Z_p^*$ for $|A| > p^{11/23 +\epsilon }$. In addition, we extend a result of Shkredov to show that $|2A|\gg |A|^{8/5-\epsilon }$ for $|A|\ll p^{5/9}$.
