On an entropic analogue of additive energy

TL;DR

This paper introduces an entropic analogue of additive energy, explores its theoretical properties, and applies it to entropy-based sumset theorems and sum-product conjectures.

Contribution

It defines and develops the theory of entropic additive energy, linking it to sumset inequalities and sum-product conjectures in finite fields.

Findings

Entropic additive energy relates to sumset cardinalities and entropy inequalities.

Demonstrates the role of entropic energy in Tao's entropy variant of the Balog--Szemerédi--Gowers theorem.

Proposes sum-product conjectures connecting entropic additive and multiplicative energies.

Abstract

Recent advances have linked various statements involving sumsets and cardinalities with corresponding statements involving sums of random variables and entropies. In this vein, this paper shows that the quantity $2{\bf H}\{X, Y\} - {\bf H}\{X+Y\}$ is a natural entropic analogue of the additive energy $E(A,B)$ between two sets. We develop some basic theory surrounding this quantity, and demonstrate its role in the proof of Tao's entropy variant of the Balog--Szemer\'edi--Gowers theorem. We examine the regime where entropic additive energy is small, and discuss a family of random variables related to Sidon sets. In finite fields, one can define an entropic multiplicative energy as well, and we formulate sum-product-type conjectures relating these two entropic energies.