Any small multiplicative subgroup is not a sumset

Abstract

We prove that for an arbitrary $\epsilon >0$ and any multiplicative subgroup $\mathrm{\Gamma }\subseteq {\mathbb{F}}_p$, $1\ll |\mathrm{\Gamma }|⩽p^{2/3-\epsilon }$ there are no sets B, $C\subseteq {\mathbb{F}}_p$ with $|B|,|C|>1$ such that $\mathrm{\Gamma }=B+C$. Also, we obtain that for $1\ll |\mathrm{\Gamma }|⩽p^{6/7-\epsilon }$ and any $\xi \ne 0$ there is no a set B such that $\xi \mathrm{\Gamma }+1=B/B$.