The Borel—Bernstein Theorem for multidimensional continued fractions

摘要

A central result in the metric theory of continued fractions, the Borel—Bernstein Theorem gives statistical information on the rate of increase of the partial quotients. We introduce a geometrical interpretation of the continued fraction algorithm; then, using this set-up, we generalize it to higher dimensions. In this manner, we can define known multidimensional algorithms such as Jacobi—Perron, Poincaré, Brun, Rauzy induction process for interval exchange transformations, etc. For the standard continued fractions, partial quotients become return times in the geometrical approach. The same definition holds for the multidimensional case. We prove that the Borel—Bernstein Theorem holds for recurrent multidimensional continued fraction algorithms.