A Note on Sumsets using Entropy

Gyarmati, Matolcsi, and Ruzsa recently noted [2] that Han-type inequalities can be applied to sumsets in much the same way that they can be applied to characteristic functions of sets of random variables (the usual situation). While it is not true (in general) that sumsets satisfy a logsubmodular relation in an obvious way, it is natural to ask whether they permit a weaker property, by way of fractional subadditivity. It is classical (and recently reviewed in [4]) that fractional subadditivity is weaker than log-submodularity and more general than Han’s inequalities. Here, we extend an argument of [2] (more precisely, an idea in the proof of Theorem 1.2 in their paper), make further use of entropy, and show general fractional subadditivity properties for sumsets that imply some of the results and conjectures in[2] as easy corollaries. It should be noted that some of the ideas in this paper were discovered independently by Balister and Bollobás [1] who, following the works of [2] and [4], also developed a hierarchy of entropy inequalities. This paper, however, constains some new ideas that can be used to extend the results in [1] beyond sumsets in ways that were not considered in that paper.