On the distribution of additive energy revisited

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.

💡 Research Summary

The paper “On the Distribution of Additive Energy Revisited” investigates two intertwined topics in the setting of a prime finite field (\mathbb{F}_p): (1) the quantitative distribution of multiplicative (and additive) energies of subsets, and (2) the covering properties of product sets when the underlying set has small additive doubling.

1. Energy definitions and basic monotonicity.
For a set (A\subset\mathbb{F}_p) the (k)-fold multiplicative energy is defined as
\