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arXiv · 2104.03405

Rational approximations to two irrational numbers

Nikita Shulga
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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.

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Nikita Shulga. 2021-04-07. Rational approximations to two irrational numbers. https://doi.org/10.2140/moscow.2022.11.1

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