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An Analytic Approach to Cardinalities of Sumsets

Abstract

Let d be a positive integer and U ⊂ ℤd finite. We study

$$\beta (U): = \mathop {\inf }\limits_{\mathop {A,B \ne \phi }\limits_{{\rm{finite}}} } {{\left| {A + B + U} \right|} \over {{{\left| A \right|}^{1/2}}{{\left| B \right|}^{1/2}}}},$$

and other related quantities. We employ tensorization, which is not available for the doubling constant, ∣U + U∣/∣U∣. For instance, we show

$$\beta (U) = \left| U \right|,$$

whenever U is a subset of {0,1}d. Our methods parallel those used for the Prékopa—Leindler inequality, an integral variant of the Brunn—Minkowski inequality.

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Acknowledgement

GS is supported by Ben Green’s Simons Investigator Grant 376201. DZ is supported by a Knut and Alice Wallenberg Fellowship (Program for Mathematics 2017). IR is supported by Hungarian National Foundation for Scientific Research (OTKA), Grants No. T 29759, T 38396 and K129335. MD is supported by the New National Excellence Program of the National Research, Development and Innovation Fund and the Ministry for Innovation and Technology of Hungary (ÚNKP-20-1). The authors thank Máté Matolcsi, Thomas Bloom, Misha Rudnev, Oliver Roche-Newton, Ben Green and Ilya Shkredov for useful discussions.

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Correspondence to George Shakan.

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Matolcsi, D., Ruzsa, I.Z., Shakan, G. et al. An Analytic Approach to Cardinalities of Sumsets. Combinatorica 42, 203–236 (2022). https://doi.org/10.1007/s00493-021-4547-0

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  • DOI: https://doi.org/10.1007/s00493-021-4547-0

Mathematics Subject Classification (2010)

  • 11B13
  • 11B75