PreprintCC BY 4.0arXiv
One Arithmetic Gadget, Two Problems
Abstract
There is an absolute constant c>0 such that, for every 0<ε≤1, arbitrarily large finite sets A⊂ℂ satisfy |A+A|≤ |A|1+ε, |AA|≤ |A|2−cε, ν1(A)≥ |A|1+cε, where ν1(A) counts unordered unit-distance pairs. We combine the arithmetic directions from the recent unit-distance construction with the multiplicative enlargement used in the recent sum-product construction. The arithmetic ingredients are stated as explicit inputs.
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BibTeX
@misc{solymosi2026arithmetic,
title = {{One Arithmetic Gadget, Two Problems}},
author = {Jozsef Solymosi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.17453},
eprint = {2609.17453},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.17453},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - One Arithmetic Gadget, Two Problems
AU - Jozsef Solymosi
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.17453
UR - https://arxiv.org/abs/2609.17453
AB - There is an absolute constant c>0 such that, for every 0<ε≤1, arbitrarily large finite sets A⊂ℂ satisfy |A+A|≤ |A|^1+ε, |AA|≤ |A|^2−cε, ν_1(A)≥ |A|^1+cε, where ν_1(A) counts unordered unit-distance pairs. We combine the arithmetic directions from the recent unit-distance construction with the multiplicative enlargement used in the recent sum-product construction. The arithmetic ingredients are stated as explicit inputs.
ER -