A note on general isolation result in Diophantine Approximation

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 $\left|\hat{I}\pm -\frac{p}{q}\right|<\frac{f(q)}{q^2}$ leads to the existence of solutions of inequality $\left|\hat{I}\pm -\frac{p}{q}\right|<\frac{f_1(q)}{q^2}$.