On the distribution of additive energy revisited
Norbert Hegyvári
TL;DR
The paper addresses the distribution of multiplicative energy values $E^ imes_k(A)$ for subsets of the finite field $\\mathbb{F}_p$, extending prior additive-energy work to the multiplicative setting. It combines Fourier-analytic techniques with random-structure methods to obtain new structural results and to construct sets showing large growth in the ratio $\frac{E^ imes_4(A)}{|A|\,E^ imes_3(A)}$, including a region in the $(\eta,\mu)$-plane where this ratio diverges as $p\to\infty$. A key methodological ingredient is the Elekes–Garaev framework paired with Kim–Vu concentration to realize sets with prescribed energy exponents and with small-doubling behavior, leading to consequences about growth of $A^3$ and subgroup containment like $A^{12K}=G$. The results advance understanding of energy distribution in finite fields and illuminate when multiplicative energy growth dominates, with implications for multiplicative expansion and structure.
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.