A note on general isolation result in Diophantine Approximation

TL;DR

The paper develops general, isolation-type statements in Diophantine approximation by relating baseline approximation bounds to stronger, infinitely often attainable bounds. It introduces a matrix-approximation framework and a structured class of admissible error functions $\mathfrak{F}$, proving a general result (Theorem 3) that guarantees a nontrivial function $g$ can be subtracted from the threshold without losing infinite solvability, with a rational-valued case refined via an explicit $g_f$. The approach hinges on a two-case analysis tied to the homogeneous-approximation sets $\mathcal{M}$ and $\mathcal{M}^c$, and culminates in a complete proof of the rational-case strengthening (Theorem 2) using continued fractions, thereby extending classical connections to Markoff/Lagrange spectra and informing the interplay of two irrationality measures.

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