TY - JOUR

T1 - On the density of sumsets

AU - Leonetti, Paolo

AU - Tringali, Salvatore

N1 - Funding Information:
P.L. was supported by the Austrian Science Fund (FWF), project F5512-N26 and by PRIN 2017, Grant 2017CY2NCA. Both authors thank the anonymous reviewers for a careful reading of the manuscript and many suggestions that helped to improve the overall quality of the paper.
Funding Information:
P.L. was supported by the Austrian Science Fund (FWF), project F5512-N26 and by PRIN 2017, Grant 2017CY2NCA.
Publisher Copyright:
© 2022, The Author(s).

PY - 2022/7

Y1 - 2022/7

N2 - 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.

AB - 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.

KW - Analytic density

KW - Asymptotic density

KW - Banach density

KW - Buck density

KW - Logarithmic density

KW - Sumsets

KW - Upper and lower densities (and quasi-densities)

UR - http://www.scopus.com/inward/record.url?scp=85126785522&partnerID=8YFLogxK

U2 - 10.1007/s00605-022-01694-1

DO - 10.1007/s00605-022-01694-1

M3 - Article

AN - SCOPUS:85126785522

VL - 198

SP - 565

EP - 580

JO - Monatshefte für Mathematik

JF - Monatshefte für Mathematik

SN - 0026-9255

IS - 3

ER -