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$B_h$-sets of real and complex numbers

Part of: Spaces with richer structures Sequences and sets

Published online by Cambridge University Press:  11 June 2025

Melvyn B. Nathanson Open the ORCID record for Melvyn B. Nathanson [Opens in a new window]
Melvyn B. Nathanson*
Affiliation:
Department of Mathematics, Lehman College (CUNY), Bronx, NY 10468, United States
*
e-mail: melvyn.nathanson@lehman.cuny.edu
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Abstract

Let $K = \mathbf {R}$ or $\mathbf {C}$. An n-element subset A of K is a $B_h$-set if every element of K has at most one representation as the sum of h not necessarily distinct elements of A. Associated with the $B_h$-set $A = \{a_1,\ldots , a_n\}$ are the $B_h$-vectors $\mathbf {a} = (a_1,\ldots , a_n)$ in $K^n$. This article proves that “almost all” n-element subsets of K are $B_h$-sets in the sense that the set of all $B_h$-vectors is a dense open subset of $K^n$.

Keywords

Sidon setBh-setopen dense subsetsBaire’s theoremadditive number theory

MSC classification

Primary: 11B05: Density, gaps, topology 11B13: Additive bases, including sumsets 11B30: Arithmetic combinatorics; higher degree uniformity 11B34: Representation functions 11B75: Other combinatorial number theory 54E52: Baire category, Baire spaces

Information

Type
Article
Information
Canadian Mathematical Bulletin , Volume 69 , Issue 1 , March 2026 , pp. 137 - 141
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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

Supported in part by PSC-CUNY Research Award Program grant 66197-00 54.

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