Entropy and set cardinality inequalities for partition‐determined functions |
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Authors: | Mokshay Madiman Prasad Tetali |
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Institution: | Department of Statistics, Yale University, New Haven, Connecticut 06511M. Madiman was supported by Yale University (Junior Faculty Fellowship), and by NSF (CAREER grant DMS 1056996 and CCF 1065494). |
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Abstract: | A new notion of partition‐determined functions is introduced, and several basic inequalities are developed for the entropies 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. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 40, 399–424, 2012 |
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Keywords: | sumsets additive combinatorics entropy inequalities cardinality inequalities |
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