Rational approximations to two irrational numbers

Abstract

For real $\xi$ we consider the irrationality measure function ${\psi }_{\xi }(t)={\min }_{1⩽q⩽t,q\in \mathbb{Z}}||q\xi ||$, 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 $$|\frac{1}{{\psi }_{\alpha }(t)}-\frac{1}{{\psi }_{\beta }(t)}|⩾\sqrt{5}\left(1-\sqrt{\frac{\sqrt{5}-1}{2}}\right)t.$$ The constant on the right-hand side is optimal.

11 pages

References in corpus (1)

• Approximations to two real numbers

Cited by in corpus (1)

• On irrationality measure functions for several real numbers