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PreprintCC BY 4.0arXiv

On a Notion of Density Related to Restricted Diophantine Approximation

  • Chance Sanford
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Abstract

We introduce notion of density for a subset of the natural numbers called the Diophantine density. First, we demonstrate that the Diophantine density is an abstract density in the sense of Di Nasso and Jin. Then we establish upper and lower bounds for the Diophantine density of a set and its complement respectively. We also exhibit a construction which allows us to find a set whose density can be any value between zero and one. Finally, we prove a restricted-denominator analog of Dirichlet's approximation theorem for sets of non-zero Diophantine density.

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Cite

BibTeXRIS
BibTeX
@misc{sanford2026notion,
  title = {{On a Notion of Density Related to Restricted Diophantine Approximation}},
  author = {Chance Sanford},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.27497},
  eprint = {2610.27497},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.27497},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - On a Notion of Density Related to Restricted Diophantine Approximation
AU  - Chance Sanford
PY  - 2026
DA  - 2026/10/20
PB  - arXiv
DO  - 10.48550/arxiv.2610.27497
UR  - https://arxiv.org/abs/2610.27497
AB  - We introduce notion of density for a subset of the natural numbers called the Diophantine density. First, we demonstrate that the Diophantine density is an abstract density in the sense of Di Nasso and Jin. Then we establish upper and lower bounds for the Diophantine density of a set and its complement respectively. We also exhibit a construction which allows us to find a set whose density can be any value between zero and one. Finally, we prove a restricted-denominator analog of Dirichlet's approximation theorem for sets of non-zero Diophantine density.
ER  -