Restricted sumsets in multiplicative subgroups

TL;DR

This work establishes a restricted-sum analogue of Sárközy’s additive-decomposition conjecture for multiplicative subgroups of finite fields, proving that the set of nonzero squares $S_2$ in $\,\mathbb{F}_q$ with odd order $q>13$ cannot be written as $A\hat{+}A$. The authors develop a robust toolkit combining Stepanov’s method with hyper-derivatives to derive sharp upper bounds on $|A|$ when $A\hat{+}A\subset S_d$, and they connect these results to Cayley sum graphs, yielding a restricted-sum version of van Lint–MacWilliams type statements. They further show that, if a restricted sumset decomposition $A\hat{+}A=S_d$ exists, $A$ is highly structured (often Sidon), and they quantify the rarity of such decompositions among primes via density arguments. The paper also extends the analysis to integers through Gallagher’s sieve, obtaining near-optimal bounds for $|A|$ in the Erdős–Moser context and illustrating broad implications for restricted-sum decompositions in combinatorics and number theory.

Abstract

We establish the restricted sumset analogue of the celebrated conjecture of Sárközy on additive decompositions of the set of nonzero squares over a finite field. More precisely, we show that if $q>13$ is an odd prime power, then the set of nonzero squares in $\mathbb{F}_q$ cannot be written as a restricted sumset $A \hat{+} A$, extending a result of Shkredov. More generally, we study restricted sumsets in multiplicative subgroups over finite fields as well as restricted sumsets in perfect powers (over integers) motivated by a question of Erdős and Moser. We also prove an analogue of van Lint-MacWilliams' conjecture for restricted sumsets, which appears to be the first analogue of Erdős-Ko-Rado theorem in a family of Cayley sum graphs.

Theorems & Definitions (40)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Corollary 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Definition 2.1
• Lemma 2.2: LN97