PreprintCC0arXiv
On a question of Ruzsa about densities of sumsets
Abstract
Let d, d∗, and d∗ denote, respectively, the asymptotic density, the upper asymptotic density, and the lower asymptotic density on ℕ. We show that there are no constants 0<ν<1 and c>0 such that d∗(A+A)≥ c(d∗(A+A))1−νd(A)ν for every set A⊆ ℕ which admits asymptotic density. This answers in the negative a question of I.Z. Ruzsa.
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BibTeX
@misc{leonetti2026question,
title = {{On a question of Ruzsa about densities of sumsets}},
author = {Paolo Leonetti},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.20206},
eprint = {2609.20206},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20206},
note = {CC0},
}
RIS
TY - UNPB
TI - On a question of Ruzsa about densities of sumsets
AU - Paolo Leonetti
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.20206
UR - https://arxiv.org/abs/2609.20206
AB - Let d, d^∗, and d_∗ denote, respectively, the asymptotic density, the upper asymptotic density, and the lower asymptotic density on ℕ. We show that there are no constants 0<ν<1 and c>0 such that d_∗(A+A)≥ c(d^∗(A+A))^1−νd(A)^ν for every set A⊆ ℕ which admits asymptotic density. This answers in the negative a question of I.Z. Ruzsa.
ER -