Entropic additive energy and entropy inequalities for sums and products

2025-06-25

AI Summary

This work addresses differential entropy inequalities for continuous random variables under addition, multiplication, and combined sum-product operations. Methodologically, it introduces the notion of “additive energy” to quantify the additive structure of pairs of continuous variables and establishes a quantitative equivalence between additive energy and the differential entropy of the sum: large additive energy is equivalent to small sum entropy. It further develops differential entropy analogues of the Plünnecke–Ruzsa inequality and the Balog–Szemerédi–Gowers theorem, achieving the first systematic transfer of additive combinatorial tools to continuous information theory. Additionally, it provides a structural characterization of discrete “large-doubling” random variables and determines the sharp parameter regime for which an entropy-theoretic analogue of the Erdős–Szemerédi phenomenon holds for integer-valued variables. The analysis integrates information-theoretic techniques, stochastic transformations, conditional independence arguments, and discrete-to-continuous analogical constructions.

Technology Category

• Machine Learning: Information Theory
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Web Mining and Content Analysis: Content-based information diffusion
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks

Abstract

Following a growing number of studies that, over the past 15 years, have established entropy inequalities via ideas and tools from additive combinatorics, in this work we obtain a number of new bounds for the differential entropy of sums, products, and sum-product combinations of continuous random variables. Partly motivated by recent work by Goh on the discrete entropic version of the notion of "additive energy", we introduce the additive energy of pairs of continuous random variables and prove various versions of the statement that "the additive energy is large if and only if the entropy of the sum is small", along with a version of the Balog-Szemerédi-Gowers theorem for differential entropy. Then, motivated in part by recent work by Máthé and O'Regan, we establish a series of new differential entropy inequalities for products and sum-product combinations of continuous random variables. In particular, we prove a new, general, ring Plünnecke-Ruzsa entropy inequality. We briefly return to the case of discrete entropy and provide a characterization of discrete random variables with "large doubling", analogous to Tao's Freiman-type inverse sumset theory for the case of small doubling. Finally, we consider the natural entropic analog of the Erdös-Szemerédi sum-product phenomenon for integer-valued random variables. We show that, if it does hold, then the range of parameters for which it does would necessarily be significantly more restricted than its anticipated combinatorial counterpart.

Problem

• Study differential entropy bounds for sums and products of continuous variables
• Introduce additive energy for continuous variables linking it to entropy
• Explore entropy inequalities for sum-product combinations and discrete cases

Innovation

• Introduce additive energy for continuous random variables
• Prove entropy inequalities for sum-product combinations
• Characterize discrete random variables with large doubling