Entropy versions of additive inequalities

TL;DR

This paper extends a method linking additive combinatorics inequalities with entropy-based versions, providing a versatile toolkit for proving known results more simply and discovering new inequalities.

Contribution

It introduces an extension of Ruzsa's device, enhancing the ability to establish equivalences between sumset and entropic inequalities in additive combinatorics.

Findings

Simplified proofs of existing inequalities

A new framework for deriving entropic inequalities

Potential to discover novel additive combinatorics results

Abstract

The connection between inequalities in additive combinatorics and analogous versions in terms of the entropy of random variables has been extensively explored over the past few years. This paper extends a device introduced by Ruzsa in his seminal work introducing this correspondence. This extension provides a toolbox for establishing the equivalence between sumset inequalities and their entropic versions. It supplies simpler proofs of known results and opens a path for obtaining new ones.