Title: Extreme Value Theorem for geodesic flow on the quotient of the theta group
Abstract: The extreme value theorem (EVT) is a theory concerning the distribution of the maximum of a sequence of random variables. For classical continued fractions, an EVT has been established, describing the limiting distribution of large partial quotients. Their significance is further explained by the correspondence with geodesic flow on the modular surface, where large partial quotients can be seen as excursions to the cusp. In this talk, we first establish an EVT for a modified version of the even continued fraction expansion. This result is then used to derive the EVT for the geodesic flow on the hyperbolic surface associated with the theta group, which has two cusps. This is joint work with Jaelin Kim and Seonhee Lim.
Title: Shrinking Target Problem for Matrix Transformations of Tori
Abstract: We study matrices with real coefficients as transformations on the d-dimensional torus. We are interested in the size of the shrinking target sets, i.e., the sets of the points whose orbits under a fixed matrix transformation fall into a family of shrinking subsets infinitely often. We prove a zero-one law for the Lebesgue measure of such sets. We also give a Hausdorff dimension formula for the diagonal matrix transformations. This is a joint work with Bing Li, Sanju Velani and Evgeniy Zorin.
Title: Weighted uniform distribution of subpolynomial functions in Hardy fields
Abstract: M. Boshernitzan established a necessary and sufficient condition for the sequence f(n) to be uniformly distributed modulo 1, where f(x) is a subpolynomial function in a Hardy field. In previous joint work with Vitaly Bergelson and Grigori Kolesnik, we extended this result by showing that f(n) is uniformly distributed modulo 1 if and only if f(p_n) is uniformly distributed modulo 1, where p_n denotes the n-th prime. In this talk, we further consider the weighted uniform distribution, inspired by earlier work of M. Tsuji. We present an extension of the previous result on uniform distribution for subpolynomials in a Hardy field and discuss some applications of this extension. This is a joint work with Vitaly Bergelson and Grigori Kolesnik.
Contact: Dong Han Kim (kim2010@dgu.ac.kr)