Schmidt's game, its modifications, and a conjecture of Margulis
Time and place
1 PM on Thursday, February 10th, 2011; NAC 6-113
Barak Weiss (Ben Gurion University, Israel)
Abstract
Let BA denote the set of real numbers with bounded continued fraction coefficients. This is a set of zero Lebesgue which is also meager (small in the sense of category). Nevertheless it was shown by W. Schmidt in 1966 that for any sequence f_1, f_2, ... of C^1 bijective maps of the reals, the countable intersection of f_i(BA) is nonempty. In proving this result Schmidt introduced a powerful (yet amusing) method based on a game for two players, which can be played on any complete metric space. Recently new variants of Schmidt's game have been devised, which make it possible to extend Schmidt's results, as well as to show that certain dynamically defined sets have nonempty intersection. As a consequence, in joint work with Dmitry Kleinbock, we verify a conjecture of Margulis from 1990.
