PreprintCC BY 4.0arXiv
Bh-sets and perturbations in normed vector spaces
Abstract
The subset A = {ai:i ∈ I} of a normed vector space is a Bh-set if every element of the sumset hA has a unique representation as a sum of h elements of A. Let ε = {εi:i ∈ I} be a set of positive real numbers. An ε-perturbation of A is a set A' = {a'i:i∈ I} such that |ai'−ai|<εi for all i ∈ I. Let ΔhA = inf{|x'−x| : x,x' ∈ hA and x≠ x'}. It is proved that if A is finite or countably infinite set with ΔhA>0, then there is a Bh-set A' that is an ε-perturbation of A.
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BibTeX
@misc{nathanson2026bhsets,
title = {{$B_h$-sets and perturbations in normed vector spaces}},
author = {Melvyn B. Nathanson},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09249},
eprint = {2609.09249},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09249},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - B_h-sets and perturbations in normed vector spaces
AU - Melvyn B. Nathanson
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.09249
UR - https://arxiv.org/abs/2609.09249
AB - The subset A = {a_i:i ∈ I} of a normed vector space is a B_h-set if every element of the sumset hA has a unique representation as a sum of h elements of A. Let ε = {ε_i:i ∈ I} be a set of positive real numbers. An ε-perturbation of A is a set A' = {a'_i:i∈ I} such that |a_i'−a_i|<ε_i for all i ∈ I. Let Δ_hA = inf{|x'−x| : x,x' ∈ hA and x≠ x'}. It is proved that if A is finite or countably infinite set with Δ_hA>0, then there is a B_h-set A' that is an ε-perturbation of A.
ER -