A note on general isolation result in Diophantine Approximation
Sergei Pitcyn, Nikolay Moshchevitin
公開日: 2025/9/30
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)>f_1(x)>0$ such that existence of solutions $\frac{p}{q}$ of Diophantine inequality $ \left| \alpha -\frac{p}{q}\right|< \frac{f(q)}{q^2} $ leads to the existence of solutions of inequality $ \left| \alpha -\frac{p}{q}\right|< \frac{f_1(q)}{q^2} $.