Rational approximations to two irrational numbers

TL;DR

This paper investigates the approximation properties of two irrational numbers, establishing a sharp inequality relating their irrationality measures for large values of t, and demonstrating the optimality of the constant involved.

Contribution

It proves a new inequality connecting the irrationality measures of two irrationals under certain conditions, with an optimal constant, advancing understanding of their approximation behavior.

Findings

Established a lower bound for the difference of reciprocals of irrationality measures.

Proved the constant in the inequality is optimal.

Applied the inequality to characterize approximation properties of irrationals.

Abstract

For real $\xi$ we consider the irrationality measure function $\psi_\xi(t) = \min_{1\leqslant q \leqslant t, q\in\mathbb{Z}} || q\xi ||$, where $||\cdot||$ - distance to the nearest integer. We prove that in the case $\alpha\pm\beta\notin\mathbb{Z}$ there exist arbitrary large values of $t$ with $$\Bigl | \frac{1}{\psi_\alpha(t)} - \frac{1}{\psi_\beta(t)} \Bigl | \geqslant \sqrt5\left(1-\sqrt{\frac{\sqrt5-1}{2}}\right)t.$$ The constant on the right-hand side is optimal.