Additive decompositions of cubes in finite fields

TL;DR

This paper investigates the additive decomposition properties of the set of non-zero cubes in finite fields, proving non-existence of certain decompositions for large primes.

Contribution

It establishes new bounds and conditions under which the set of non-zero cubes cannot be decomposed into sums of smaller subsets in finite fields.

Findings

No decomposition of the form C_p=A+B+C exists for primes p > 184291 with |A|,|B|,|C| ≥ 2

Provides bounds on additive decompositions of cube sets in finite fields

Enhances understanding of additive structures in finite fields

Abstract

Let $p\equiv1\pmod3$ be a prime . We study several topics on additive decompositions concerning the set $C_p$ of all non-zero cubes in the finite field of $p$ elements. For example, we show that when $p>184291$ , the set $C_p$ has no decomposition of the form $C_p=A+B+C$ with $|A|,|B|,|C|\ge2$.