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Explicit sumset sizes in additive number theory

Part of: Sequences and sets

Published online by Cambridge University Press:  04 December 2025

Melvyn B. Nathanson Open the ORCID record for Melvyn B. Nathanson [Opens in a new window]
Melvyn B. Nathanson*
Affiliation:
Lehman College of the City University of New York , USA
*
e-mail: melvyn.nathanson@lehman.cuny.edu
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Abstract

It is an open problem in additive number theory to compute and understand the full range of sumset sizes of finite sets of integers, that is, the set $ \mathcal R_{\mathbf Z}(h,k) = \{|hA|:A \subseteq \mathbf Z \text { and } |A|=k\}$ for all integers $h \geq 3$ and $k \geq 3$. This article constructs certain infinite families of finite sets of size k, computes their h-fold sumset sizes, and obtains explicit finite arithmetic progressions of sumset sizes in $ \mathcal R_{\mathbf Z}(h,k)$.

Keywords

Sumsetsumset sizesadditive number theory

MSC classification

Primary: 11B05: Density, gaps, topology 11B30: Arithmetic combinatorics; higher degree uniformity 11B75: Other combinatorial number theory

Information

Type
Article
Information
Canadian Mathematical Bulletin , First View , pp. 1 - 12
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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Footnotes

This work was supported in part by the PSC-CUNY Research Award Program grant 66197-00 54.

References

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