Any small multiplicative sugroup is not a sumset

TL;DR

The paper proves that small multiplicative subgroups in finite fields cannot be expressed as sumsets or certain ratio sets, revealing structural limitations of these subgroups.

Contribution

It establishes new bounds showing that small multiplicative subgroups cannot be represented as sumsets or specific ratio sets, advancing understanding of their additive structure.

Findings

No sumset representation for small subgroups with size up to p^{2/3 - ε}.

No ratio set representation for subgroups up to size p^{6/7 - ε}.

Results improve bounds on additive properties of multiplicative subgroups.

Abstract

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