On the distribution of additive energy revisited

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.