On the distribution of additive energy revisited

TL;DR

This paper investigates the distribution of additive and multiplicative energies in finite fields using Fourier analysis, offering new structural insights and bounds related to product sets and small doubling properties.

Contribution

It introduces novel Fourier analytical methods to analyze energy distribution and provides bounds for the minimal product set covering the entire field.

Findings

New bounds for energy distribution in finite fields

Structural insights into energy phenomena with small doubling sets

Estimate for the smallest k such that A^k covers the entire field

Abstract

This paper extends the investigation of energy distribution in finite settings, which is related to the results established in [H]. We analyze the distribution of multiplicative energies using Fourier analytical methods and random structures. Our results provide new structural insights into energy phenomena in finite fields, complementing the earlier discrete analysis. Additionally, we provide an estimate for the smallest $k$ such that the $k$-fold product set $A^k$ covers the entire field $\mathbb{F}$, given that $A$ has small doubling.