On sumsets involving $k$th powers of finite fields

TL;DR

The paper analyzes additive decompositions of the set $D_k$ of $k$th power residues in $\mathbb{F}_p$ for primes $p\equiv1\pmod{k}$. Employing the polynomial method, Weil-type character-sum bounds, and divisor-density techniques (Ford), it obtains quantitative size constraints for two-term decompositions $D_k=A+B$ (with $|A||B|=|D_k|=(p-1)/k$), including explicit lower bounds $|A|,|B|\ge\frac{\sqrt{p}}{6k}$ for certain ranges of $p$ and asymptotic bounds $|A|,|B|\approx c\sqrt{p}$ as $p$ grows. It then rules out nontrivial three-term decompositions $D_k=A+B+C$ for large $p$ via a Ruzsa-type inequality, showing no such decomposition is possible when $p$ is sufficiently large. Finally, it proves that the set of primes $p\equiv1\pmod{k}$ for which $D_k$ admits a nontrivial $2$-additive decomposition has density $0$, with an explicit sparse-bound in terms of divisor-counting functions. The results extend prior work on quadratic residues to $k$th powers, contributing quantitative obstructions to additive decompositions in finite fields and advancing the understanding of sumsets in additive combinatorics.

Abstract

In this paper, we study some topics concerning the additive decompositions of the set $D_k$ of all $k$th power residues modulo a prime $p$. For example, given a positive integer $k\ge2$, we prove that $$\lim_{x\rightarrow+\infty}\frac{B(x)}{π(x)}=0,$$ where $π(x)$ is the number of primes $p\le x$ and $B(x)$ denotes the cardinality of the set $$\{p\le x: p\equiv1\pmod k; D_k\ \text{has a non-trivial 2-additive decomposition}\}.$$

Theorems & Definitions (11)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• Lemma 3.1