Subset-Sum Density Realization in Locally Finite Abelian Groups
Abstract
Let G be a countable locally finite abelian group, and let G1≤ G2≤⋯, ⋃i≥1Gi=G, be any filtration of G by finite subgroups. For A⊆ G, let 𝒫(A) denote the set of all finite subset sums of elements of A, and let 2G:={2g:g∈ G}. We prove that |2G|=∞ if and only if for every filtration and every interval [α,β]⊆[0,1], there exists A⊆ G such that the set of limit points of ( (|𝒫(A)∩ Gi|)/|Gi| )i≥1 is exactly [α,β]. We also prove a positive-density result that does not require |2G|=∞: if G is infinite and 𝒫(A) has positive upper density along the given filtration, then there is an infinite set B⊆ 𝒫(A) such that B+B⊆ 𝒫(A). Combining this with the realization theorem, we show that, when |2G|=∞, every interval [α,β] with 0≤α≤β≤1 and β>0 can be realized by a set A with this additional property.
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BibTeX
@misc{hegyvari2026subsetsum,
title = {{Subset-Sum Density Realization in Locally Finite Abelian Groups}},
author = {Norbert Hegyvári and Thang Pham and Boqing Xue},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14630},
eprint = {2609.14630},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14630},
}
RIS
TY - UNPB
TI - Subset-Sum Density Realization in Locally Finite Abelian Groups
AU - Norbert Hegyvári
AU - Thang Pham
AU - Boqing Xue
PY - 2026
DA - 2026/09/13
PB - arXiv
DO - 10.48550/arxiv.2609.14630
UR - https://arxiv.org/abs/2609.14630
AB - Let G be a countable locally finite abelian group, and let G_1≤ G_2≤⋯, ⋃_i≥1G_i=G, be any filtration of G by finite subgroups. For A⊆ G, let 𝒫(A) denote the set of all finite subset sums of elements of A, and let 2G:={2g:g∈ G}. We prove that |2G|=∞ if and only if for every filtration and every interval [α,β]⊆[0,1], there exists A⊆ G such that the set of limit points of ( (|𝒫(A)∩ G_i|)/|G_i| )_i≥1 is exactly [α,β]. We also prove a positive-density result that does not require |2G|=∞: if G is infinite and 𝒫(A) has positive upper density along the given filtration, then there is an infinite set B⊆ 𝒫(A) such that B+B⊆ 𝒫(A). Combining this with the realization theorem, we show that, when |2G|=∞, every interval [α,β] with 0≤α≤β≤1 and β>0 can be realized by a set A with this additional property.
ER -