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arXiv · 1702.01197

Any small multiplicative sugroup is not a sumset

Ilya D. Shkredov
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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 $\xi \Gamma+1=B/B$.

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Ilya D. Shkredov. 2017-02-03. Any small multiplicative sugroup is not a sumset. https://arxiv.org/abs/1702.01197

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