**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.