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arXiv · 2601.12457

Structure theory of set addition with two operations

Aliaksei Semchankau·Ilya Shkredov
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Abstract

We take the first step toward a structure theory that includes both operations of a ring $\mathcal{R}$. More precisely, we prove a series of inverse results for the structure of sets $A\subseteq \mathbf{F}_p$ such that, under certain conditions on integers $r_1, \dots, r_k$, one has $|A^{r_1} + \dots + A^{r_k}| \ll \sqrt[k]{p^{k-1} |A|}$.

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Aliaksei Semchankau, Ilya Shkredov. 2026-01-18. Structure theory of set addition with two operations. https://arxiv.org/abs/2601.12457

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