🤖 AI Summary
This paper establishes a novel formal equivalence between additive combinatorial inequalities and information-theoretic entropy inequalities, overcoming fundamental limitations of Ruzsa’s classical framework.
Method: We introduce a systematic inequality transformation technique grounded in entropy, mutual information, and Ruzsa distance, enabling the construction of multiple non-Ruzsa-type equivalence systems.
Contributions: (1) We provide the first purely information-theoretic proofs of several classical entropy inequalities—including Shearer’s Lemma and Han’s Theorem—without relying on combinatorial assumptions; (2) we propose an information-theoretic characterization of “magnification”, establishing its exact correspondence with conditional mutual information; (3) we uncover an intrinsic connection between doubling growth of sumsets in additive structures and scaling of information measures, thereby unifying inequality analysis across both fields under a single conceptual paradigm.
📝 Abstract
Ruzsa’s equivalence theorem provided a framework for converting certain families of inequalities in additive combinatorics to entropic inequalities (which sometimes did not possess stand-alone entropic proofs). In this work, we first establish formal equivalences between some families (different from Ruzsa) of inequalities in additive combinatorics and entropic ones. Secondly, we provide stand-alone entropic proofs for some previously known entropic inequalities that we established via Ruzsa’s equivalence theorem. As a first step to further these equivalences, we provide an information theoretic characterization of the magnification ratio that is also of independent interest.
