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On the density of A+B when B has few elements
Abstract
Let d be the asymptotic density on the positive integers ℕ+. We provide sufficient conditions on an infinite set B⊆ ℕ+ so that for each α∈ [0,1] there exists a set A⊆ ℕ+ such that d(A)=0 and d(A+B)=α. For instance, if (bn: n ∈ ℕ+) is the increasing enumeration of B, it is sufficient that liminfn bn+1/bn>1. Moreover, we show that, if |B∩ [1,n]|≪ log(n)1−ε for some ε>0, then for each α∈ [0,1] there exists a set A⊆ ℕ+ such that d(A)=d(A+B)=α. Conversely, we show that, for each ε>0, there exists an infinite set B⊆ ℕ+ such that |B∩ [1,n]| ≪ n2/3+ε and d(A+B) ∈ {0,1} for every A⊆ ℕ+ such that A+B admits asymptotic density.
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BibTeX
@misc{leonetti2026density,
title = {{On the density of $A+B$ when $B$ has few elements}},
author = {Paolo Leonetti},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29660},
eprint = {2609.29660},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29660},
}
RIS
TY - UNPB
TI - On the density of A+B when B has few elements
AU - Paolo Leonetti
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.29660
UR - https://arxiv.org/abs/2609.29660
AB - Let d be the asymptotic density on the positive integers ℕ^+. We provide sufficient conditions on an infinite set B⊆ ℕ^+ so that for each α∈ [0,1] there exists a set A⊆ ℕ^+ such that d(A)=0 and d(A+B)=α. For instance, if (b_n: n ∈ ℕ^+) is the increasing enumeration of B, it is sufficient that liminf_n b_n+1/b_n>1. Moreover, we show that, if |B∩ [1,n]|≪ log(n)^1−ε for some ε>0, then for each α∈ [0,1] there exists a set A⊆ ℕ^+ such that d(A)=d(A+B)=α. Conversely, we show that, for each ε>0, there exists an infinite set B⊆ ℕ^+ such that |B∩ [1,n]| ≪ n^2/3+ε and d(A+B) ∈ {0,1} for every A⊆ ℕ^+ such that A+B admits asymptotic density.
ER -