Any small multiplicative sugroup is not a sumset

Abstract

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

13 pages

References in corpus (3)

• Some new inequalities in additive combinatorics
• Quadratic residues and difference sets
• Collinear triples and quadruples for Cartesian products in ${\mathbb{F}}_p^2$