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 a_1, a_2, ... of reals, the countable
intersection of BA + a_i 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. In recent work
with Dmitry Kleinbock we describe variants of Schmidt's game which make it
possible to show that certain dynamically defined sets have nonempty
intersection. As a consequence we verify a conjecture of Margulis from 1990.
