Entropy and set cardinality inequalities for partition-determined functions

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.

💡 Research Summary

The paper introduces a new class of multivariate functions called partition‑determined functions and develops a unified framework that yields entropy inequalities for such functions of independent random variables as well as cardinality inequalities for the corresponding compound sets. A function (f(x_{1},\dots ,x_{k})) is said to be partition‑determined if there exists a collection of index subsets (S_{1},\dots ,S_{m}\subseteq{1,\dots ,k}) such that the value of (f) is uniquely fixed once the values of the arguments belonging to any one of these subsets are known. Classical examples include the sum map ((x,y)\mapsto x+y) in an abelian group, the product map in a (possibly non‑abelian) group, and more elaborate algebraic expressions that can be decomposed into “local” pieces.

The first major contribution is a set of entropy inequalities that generalize Shearer’s inequality and the sub‑modularity of entropy to the partition‑determined setting. For independent random variables (X_{1},\dots ,X_{k}) and a partition‑determined (f), the authors prove \