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 established via Ruzsa's equivalence theorem. As a first step to further these equivalences, we establish an information-theoretic characterization of the magnification ratio that could also be of independent interest.

翻译：Ruzsa等价定理提供了一个框架，用于将加性组合学中的某些不等式族转化为熵不等式（这些不等式有时缺乏独立的熵证明）。在本工作中，我们首先建立了加性组合学中某些（不同于Ruzsa的）不等式族与熵不等式之间的形式等价性。其次，我们为一些先前通过Ruzsa等价定理建立的已知熵不等式提供了独立的熵证明。作为推进这些等价性的第一步，我们建立了放大率的信息论刻画，该结果也可能具有独立的研究价值。