Date: 11 February 2025
Speaker: Han Zhang
Title: Khintchine’s theorem on self-similar measures on the real line
Abstract: In 1984, Mahler proposed the following question on Diophantine approximation : How close can irrational numbers in the middle-thirds Cantor set be approximated by rational numbers? One way to reformulate Mahler’s question is to ask if Khintchine’s theorem extends to the middle-thirds Cantor set. In a joint work with Timothée Bénard and Weikun He, we prove that Khintchine’s theorem holds for any self-similar measures on the real line. In particular this applies to the Hausdorff measure on the middle-thirds Cantor set. Our result generalizes the recent breakthrough work of Khalil-Luethi in dimension one. Our proof is inspired by the work of Bénard-He regarding the semisimple random walks on homogeneous spaces.
Date: 4 March 2025
Speaker: JinCheng Wang
Title: Some geometric and dynamical properties of hyperbolic magnetic flows
Abstract: As a generalization of geodesic flows, magnetic flows trace unit-speed curves with constant geodesic curvature. We consider the magnetic flows of surfaces with negative Gaussian curvature that are nonuniformly hyperbolic. By studying its geometry on the universal covering of the surface, we show the uniqueness of the measure of maximal entropy via the Bowen-Climenhaga-Thompson machinery. This is a joint work with Boris Hasselblatt.
Date: 11 March 2025
Speaker: Michael Bersudsky
Title: Equidistribution of polynomially bounded o-minimal curves in homogenous spaces
Abstract: I will present my recent joint work with Nimish Shah and Hao Xing. We study a basic question about the limiting distribution of averages along unbounded curves in homogeneous spaces. We consider the class of curves which are definable in o-minimal structures, for example, matrix curves in SL(n,R) who’s entries are rational functions.
Our results extend Ratner’s equidistribution theorem for one-parameter unipotent flows, generalize Shah’s results for polynomial curves and generalize some of the recent results Peterzil and Starchenko. A key component in our work is the Kleinbock-Margulis $(C,\alpha)$-good property for families of functions definable in polynomially bounded o-minimal structures. The property for such functions was unknown before, and its proof relies on the tools of o-minimal structures theory.
Date: 18 March 2025
Speaker: Alon Agin
Title: The Dirichlet spectrum
Abstract: Akhunzhanov and Shatskov defined the Dirichlet spectrum, corresponding to mxn matrices and to norms on R^m and R^n. In case (m,n) = (2,1) and using the Euclidean norm on R^2, they showed that the spectrum is an interval. We generalize this result to arbitrary (m,n) with max(m,n)>1 and arbitrary norms, improving previous works from recent years. We also define some related spectra and show that they too are intervals. We also prove the existence of matrices exhibiting special properties with respect to their uniform exponent. Our argument is a modification of an argument of Khintchine from 1926.
