A short note on the multiplicative energy of the spectrum of a set

TL;DR

This paper establishes an optimal upper bound for the multiplicative energy of the spectrum of any set in a finite field, advancing understanding of additive and multiplicative structures in such sets.

Contribution

It provides the first sharp upper bound for the multiplicative energy of spectra in finite fields, improving upon previous bounds related to exponential sums.

Findings

Derived the best possible upper bound for multiplicative energy.

Connected bounds to exponential sum results over subgroups.

Enhanced understanding of spectral properties in finite fields.

Abstract

We obtain an upper bound for the multiplicative energy of the spectrum of an arbitrary set from $\mathbb{F}_p$, which is the best possible up to the results on exponential sums over subgroups.