On the density of sumsets

Paolo Leonetti*, Salvatore Tringali

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


Recently introduced by the authors in [Proc. Edinb. Math. Soc. 60 (2020), 139–167], quasi-densities form a large family of real-valued functions partially defined on the power set of the integers that serve as a unifying framework for the study of many known densities (including the asymptotic density, the Banach density, the logarithmic density, the analytic density, and the Pólya density). We further contribute to this line of research by proving that (1) for each n∈ N+ and α∈ [0 , 1] , there is A⊆ N with kA∈ dom (μ) and μ(kA) = αk/ n for every quasi-density μ and every k= 1 , … , n, where kA: = A+ ⋯ + A is the k-fold sumset of A and dom (μ) denotes the domain of definition of μ; (2) for each α∈ [0 , 1] and every non-empty finite B⊆ N, there is A⊆ N with A+ B∈ dom (μ) and μ(A+ B) = α for every quasi-density μ; (3) for each α∈ [0 , 1] , there exists A⊆ N with 2 A= N such that A∈ dom (μ) and μ(A) = α for every quasi-density μ. Proofs rely on the properties of a little known density first considered by R. C. Buck and the “structure” of the set of all quasi-densities; in particular, they are rather different than previously known proofs of special cases of the same results.

Original languageEnglish
Pages (from-to)565-580
Number of pages16
JournalMonatshefte fur Mathematik
Issue number3
Publication statusPublished - Jul 2022


  • Analytic density
  • Asymptotic density
  • Banach density
  • Buck density
  • Logarithmic density
  • Sumsets
  • Upper and lower densities (and quasi-densities)

ASJC Scopus subject areas

  • Mathematics(all)


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