Entropy and set cardinality inequalities for partition-determined functions

TL;DR

This paper introduces partition-determined functions and develops entropy and set cardinality inequalities, extending classical sumset inequalities and making progress on Ruzsa's conjecture in nonabelian groups.

Contribution

It defines partition-determined functions and derives new entropy and cardinality inequalities, generalizing known sumset inequalities and advancing understanding in nonabelian group sumsets.

Findings

Entropy inequalities imply Plünnecke-Ruzsa type inequalities.

Cardinality inequalities generalize results by Gyarmati, Matolcsi, and Ruzsa.

Partial progress on Ruzsa's conjecture for nonabelian groups.

Abstract

A new notion of partition-determined functions is introduced, and several basic inequalities are developed for the entropy of such functions of independent random variables, as well as for cardinalities of compound sets obtained using these functions. Here a compound set means a set obtained by varying each argument of a function of several variables over a set associated with that argument, where all the sets are subsets of an appropriate algebraic structure so that the function is well defined. On the one hand, the entropy inequalities developed for partition-determined functions imply entropic analogues of general inequalities of Pl\"unnecke-Ruzsa type. On the other hand, the cardinality inequalities developed for compound sets imply several inequalities for sumsets, including for instance a generalization of inequalities proved by Gyarmati, Matolcsi and Ruzsa (2010). We also provide partial progress towards a conjecture of Ruzsa (2007) for sumsets in nonabelian groups. All proofs are elementary and rely on properly developing certain information-theoretic inequalities.