English

A note on general isolation result in Diophantine Approximation

Number Theory 2025-10-16 v3

Abstract

In the present paper we give very simple general statements which deal with approximation of a real number by rationals and are related to isolation phenomenon. In particular we study functions f(x)>f1(x)>0 f(x)>f_1(x)>0 such that existence of solutions pq\frac{p}{q} of Diophantine inequality αpq<f(q)q2 \left| \alpha -\frac{p}{q}\right|< \frac{f(q)}{q^2} leads to the existence of solutions of inequality αpq<f1(q)q2 \left| \alpha -\frac{p}{q}\right|< \frac{f_1(q)}{q^2} .

Keywords

Cite

@article{arxiv.2509.25628,
  title  = {A note on general isolation result in Diophantine Approximation},
  author = {Sergei Pitcyn and Nikolay Moshchevitin},
  journal= {arXiv preprint arXiv:2509.25628},
  year   = {2025}
}
R2 v1 2026-07-01T06:06:31.530Z