ИСТИНА

\usepackage[cp866]{inputenc} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsthm} \def \le {\leqslant} \def \ge {\geqslant} \topmargin -2cm \advance\voffset by 0in \oddsidemargin -1cm \hoffset 0cm \textheight = 24.5cm \textwidth 18.0cm \columnsep 10pt \columnseprule 0pt \textheight = 24.5cm \textwidth 18.0cm \columnsep 10pt \columnseprule 0pt \font\got = eufm10 scaled \magstep2 \author{ N.G. Moshchevitin} \title{ On some open problems related to simultaneous Diophantine approximation} \date{} \begin{document} \maketitle We shall discuss several problems related to simultaneous Diophantine approximation. Let $\alpha_1,...\alpha_n$ be real numbers. The object of our study is the function $$ \psi(t) =\min_{q\le t, q\in \mathbb{Z}}\,\, \max_{1\le j \le n} ||q\alpha_j|| $$ (here $||\cdot ||$ stands for the distance to the nearest integer). In particular we consider $\bullet$ problems of evaluating Diophantine constants, $\bullet$ rate of growth of denominators of best approximations, $\bullet$ inequalities between Diophantine exponents. \end{document}