Rational approximations to two irrational numbers
نویسندگان
چکیده
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 nearest integer. We prove that in 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 right-hand side is optimal.
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ژورنال
عنوان ژورنال: Moscow journal of combinatorics and number theory
سال: 2022
ISSN: ['2640-7361', '2220-5438']
DOI: https://doi.org/10.2140/moscow.2022.11.1