Skip to main content

An Analytic Approach to Cardinalities of Sumsets


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.

This is a preview of subscription content, access via your institution.


  1. J. Bourgain and M.-C. Chang: On the size of k-fold sum and product sets of integers, Journal of the American Mathematical Society 17 (2004), 473–497.

    MathSciNet  Article  Google Scholar 

  2. J. Bourgain, S. J. Dilworth, K. Ford, S. Konyagin, D. Kutzarova: Explicit constructions of RIP matrices and related problems, Duke Mathematical Journal 159 (2011), 145–185.

    MathSciNet  Article  Google Scholar 

  3. W. Beckner: Inequalities in Fourier analysis. PhD thesis, Princeton., 1975.

  4. R. Gardner: The Brunn-Minkowski inequality, Bulletin of the American Mathematical Society 39 (2002), 355–405.

    MathSciNet  Article  Google Scholar 

  5. R. J. Gardner and P. Gronchi: A Brunn-Minkowski inequality for the integer lattice, Transactions of the American Mathematical Society 353 (2001), 3995–4025.

    MathSciNet  Article  Google Scholar 

  6. B. Green, D. Matolcsi, I. Ruzsa, G. Shakan and D. Zhelezov: A weighted Prékopa—Leindler Inequality and sumsets with quasicubes, preprint.

  7. B. Green and T. Tao: Compressions, convex geometry and the Freiman—Bilu theorem, Quarterly Journal of Mathematics 57 (2006), 495–504.

    MathSciNet  Article  Google Scholar 

  8. M. Hernández Cifre, D. Iglesias and J. Yepes Nicolás: On a Discrete Brunn—Minkowski type inequality, Siam Journal of Discrete Matheamatics (2018), 1840–1856.

  9. B. Hanson, O. Roche-Newton and D. Zhelezov: On iterated product sets with shifts, II, Algebra & Number Theory 14 (2020), 2239–2260.

    MathSciNet  Article  Google Scholar 

  10. B. Halikias, D. Klartag and B. Slomka: Discrete variants of Brunn—Minkowski type inequalities, preprint arXiv:1911.040392v2, 2019.

  11. A. Iglesias, D. Nicolás and J. Zvavitch: Brunn—Minkowski type inequalities for the lattice point enumerator, preprint arXiv:1911.12874, 2019.

  12. B. Klartag and J. Lehec: Poisson processes and a log-concave Bernstein theorem, Studia Math, 2019.

  13. G. Petridis: New proofs of Plünnecke-type estimates for product sets in groups, Combinatorica 32 (2012), 721–733.

    MathSciNet  Article  Google Scholar 

  14. A. Prékopa: Logarithmic concave measures with application to stochastic programming, Acta Scientiarum Mathematicarum 32 (1971), 301–316.

    MathSciNet  MATH  Google Scholar 

  15. D. Pálvölgyi and D. Zhelezov: Query complexity and the polynomial Freiman-Ruzsa conjecture, preprint arXiv:2003.04648, 2020.

  16. G. Shakan: Nice proof of inequality (1 − xp)1/p(1 − xq)1/q ≥ (1 − x)(1 + xc)1/c where 21/c = p1/pq1/q?MathOverflow, 2019, (version: 2019-10-08).

Download references


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.

Author information

Authors and Affiliations


Corresponding author

Correspondence to George Shakan.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Matolcsi, D., Ruzsa, I.Z., Shakan, G. et al. An Analytic Approach to Cardinalities of Sumsets. Combinatorica 42, 203–236 (2022).

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI:

Mathematics Subject Classification (2010)

  • 11B13
  • 11B75