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Approximations to two real numbers

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Author(s): Igor D. Kan | Nikolay G. Moshchevitin

Journal: Uniform Distribution Theory
ISSN 1336-913X

Volume: 5;
Issue: 2;
Start page: 79;
Date: 2010;
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Keywords: Continued fraction expansionContinued fraction expansion | simultaneous approximation | badly approximateable number simultaneous approximation | badly approximateable number

ABSTRACT
For a real $\xi $put $\psi_\xi (t) = \min_{1 \le x \le t}|| x \xi ||$. Let $\alpha , \beta$ be real numbers such that $\alpha \pm \beta \not \in \mathbb{Z}$. We prove that the function $\psi_\alpha (t)-\psi_\beta (t)$ changes its sign infinitely many often as $t \to + \infty$. The proof uses continued fractions.
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