Structure theory of set addition with two operations

TL;DR

The paper develops a structure theory for sets in $\mathbb{F}_p$ under two ring-like operations by proving inverse results for small sumsets of the form $|A^{r_1}+\cdots + A^{r_k}|$. Central tools include Wiener-norm-based algebraic-intersection properties, wrappers to approximate sets by low Wiener-norm objects, and Green’s arithmetic-regularity techniques to obtain near-complete (99%) and full (100%) structural characterizations when the exponents are generic, coprime, and bounded. The results show that, under small sumset assumptions, dense sets $A$ must resemble intersections of algebraic transforms of structured sets $P_i$, with controlled Wiener norms and small additive growth, providing a robust framework for simultaneous additive and multiplicative structure in $\mathbb{F}_p$. The work advances additive combinatorics into a two-operation regime, with quantifiable structure theorems and pseudorandom-intersection properties that could influence future investigations of sum-product phenomena and ring-structure in finite fields.

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 $|A^{r_1} + \dots + A^{r_k}| \ll \sqrt[k]{p^{k-1} |A|}$.

Theorems & Definitions (53)

• Conjecture 1
• Definition 1
• Theorem 1: Lower Bound
• Theorem 2: $99\%$ Characterization of $A$
• Definition 2
• Theorem 3: $100\%$ Characterization of $A$
• Theorem : Weil
• Theorem : Bombieri
• Corollary 1
• proof