划分确定函数的熵和集合基数不等式
Entropy and set cardinality inequalities for partition-determined functions
摘要：
引入了划分决定函数的新概念，并给出了独立随机变量的划分决定函数的熵以及由划分决定函数得到的复合集的基数的几个基本不等式。这里的复合集合是指通过在与该参数相关联的集合上改变多个变量的函数的每个参数而获得的集合，其中所有集合都是适当代数结构的子集，使得函数被很好地定义。一方面，为划分确定函数发展的熵不等式意味着Plünnecke-Ruzsa型一般不等式的熵类似物。另一方面，复合集的基数不等式隐含了几个和集的不等式，包括Gyarmati，Matolcsi和Ruzsa（2010）证明的不等式的推广。我们还提供了部分进展的猜想Ruzsa（2007）的和集在nonabel群。所有的证明都是基本的，并且依赖于适当地发展某些信息论不等式。
英文摘要：
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 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.
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