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The cardinality of sumsets: different summands

Volume 167 / 2015

Brendan Murphy, Eyvindur Ari Palsson, Giorgis Petridis Acta Arithmetica 167 (2015), 375-395 MSC: Primary 11P99; Secondary 11B30. DOI: 10.4064/aa167-4-4

Abstract

We offer a complete answer to the following question on the growth of sumsets in commutative groups. Let $h$ be a positive integer and $A, B_1, \dots , B_h$ be finite sets in a commutative group. We bound $|A+B_1+\dots +B_h|$ from above in terms of $|A|$, $|A+B_1|, \dots ,|A+B_h|$ and $h$. Extremal examples, which demonstrate that the bound is asymptotically sharp in all parameters, are furthermore provided.

Authors

  • Brendan MurphyDepartment of Mathematics
    University of Rochester
    Rochester, NY 14627, U.S.A.
    e-mail
  • Eyvindur Ari PalssonDepartment of Mathematics and Statistics
    Williams College
    Williamstown, MA 01267, U.S.A.
    e-mail
  • Giorgis PetridisDepartment of Mathematics
    University of Rochester
    Rochester, NY 14627, U.S.A.
    e-mail

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