Entropy and set cardinality inequalities for partition-determined functions

Mokshay Madiman, Adam W. Marcus, Prasad Tetali

Research output: Contribution to journalArticlepeer-review

31 Scopus citations

Abstract

A new notion of partition-determined functions is introduced, and several basic inequalities are developed for the entropies 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 Plünnecke-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.

Original languageEnglish (US)
Pages (from-to)399-424
Number of pages26
JournalRandom Structures and Algorithms
Volume40
Issue number4
DOIs
StatePublished - Jul 1 2012
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Software
  • Mathematics(all)
  • Computer Graphics and Computer-Aided Design
  • Applied Mathematics

Keywords

  • Additive combinatorics
  • Cardinality inequalities
  • Entropy inequalities
  • Sumsets

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