%0 Journal Article
%T Approximations to two real numbers
%A Igor D. Kan
%A Nikolay G. Moshchevitin
%J Uniform Distribution Theory
%D 2010
%I Mathematical Institute of the Slovak Academy of Sciences
%X 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.
%K Continued fraction expansionContinued fraction expansion
%K simultaneous approximation
%K badly approximateable number simultaneous approximation
%K badly approximateable number
%U http://www.boku.ac.at/MATH/udt/vol05/no2/4KanMosh10-2.pdf