Fall 2024 Titles and Abstracts

Date: 24 September 2024
Speaker: Sam Chow
Title: Smooth discrepancy and Littlewood’s conjecture
Abstract: Given , we estimate the smooth discrepancy of the Kronecker sequence . We find that it can be smaller than the classical discrepancy of any sequence when , and can even be bounded in the case . To achieve this, we establish a novel deterministic analogue of Beck’s local-to-global principle (Annals 1994), which relates the discrepancy of a Kronecker sequence to multiplicative diophantine approximation. This opens up a new avenue of attack for Littlewood’s conjecture.

Date: 8 October 2024
Speaker: Manuel Hauke
Title: Metric Diophantine approximation: Moving targets and inhomogeneous variants
Abstract: Khintchine’s Theorem and its inhomogeneous and multidimensional variants provide a satisfying answer about the quality of approximations for almost every number. In this talk, I will discuss the (still open) question of allowing a moving target (that is, the inhomogeneous parameter changes for each denominator) in Khintchine’s Theorem. Furthermore, I will describe Duffin–Schaeffer-type results and conjectures in these setups, both in dimension 1, but also in higher dimensions. This is partially joint work with Victor Beresnevich and Sanju Velani, respectively with Felipe Ramírez.

Date: 5 November 2024
Speaker: Shreyasi Datta
Title: Fourier Asymptotics and Effective Equidistribution
Abstract: We talk about effective equidistribution of the expanding horocycles on the unit cotangent bundle of the modular surface with respect to various classes of Borel probability measures on the reals, depending on their Fourier asymptotics.  This is a joint work with Subhajit Jana.

Date: 19 November 2024
Speaker: Keivan Mallahi-Karai
Title: Spectral independence of compact groups
Abstract: Let $G_1$ and $G_2$ be  compact simple (real or $p$-adic) Lie groups, and let $\mu_1$ and $\mu_2$ be symmetric probability measures on $G_1$ and $G_2$. Under mild conditions on $\mu_1$ and $\mu_2$, the distribution of $\mu_i$ random walks on $G_i$  converges to the uniform measure, and the speed of convergence is governed by the spectral gap. A coupling of $\mu_1$ and $\mu_2$ is any probability measure $\mu$ on $G_1 \times G_2$  whose  marginal distributions  are $\mu_1$ and $\mu_2$, respectively . A natural question is under what conditions a spectral gap for all couplings depending on spectral gaps of $\mu_1$ and $\mu_2$ can be established. 
In this talk, I will present results in this direction which are based on joint work with Alireza S. Golsefidy and Amir Mohammadi. 

Date: 3 December 2024
Speaker: Gaurav Aggarwal
Title: Singular matrices on fractals
Abstract: Singular vectors are those for which Dirichlet’s Theorem can be improved by arbitrarily small multiplicative constants. Recently, Kleinbock and Weiss showed that the set of singular vectors has measure zero with respect to any friendly measure. However, determining their Hausdorff dimension remains a subtle and challenging problem. Khalil addressed this by proving that the Hausdorff dimension of the set of singular vectors intersecting a self-similar fractal is strictly smaller than the fractal’s dimension.
In this talk, I will extend Khalil’s result in four key directions. First, we generalize the study from vectors to matrices. Second, we analyze intersections with products of fractals, such as the Cartesian product of the middle-third and middle-fifth Cantor sets. Third, we establish upper bounds for singular vectors in a generalized weighted setting. Finally, we derive an upper bound on the Hausdorff dimension of $\omega$-very singular matrices in these broader settings, extending earlier work of Das, Fishman, Simmons, and Urbanski, who studied the real, unweighted case.
Our approach is dynamical in nature, relying on the construction of a height function inspired by the work of Kadyrov, Kleinbock, Lindenstrauss, and Margulis. This is a joint work with Anish Ghosh.

Date: 10 December 2024
Speaker: Noy Soffer Aranov
Title: Escape of Mass of Sequences
Abstract: One way to study the distribution of nested quadratic number fields satisfying fixed arithmetic relationships is through the evolution of continued fraction expansions. In the function field setting, it was shown by de Mathan and Teullie that given a quadratic irrational $\Theta$, the degrees of the periodic part of the continued fraction of $t^n\Theta$ are unbounded. Paulin and Shapira improved this by proving that quadratic irrationals exhibit partial escape of mass. Moreover, they conjectured that they must exhibit full escape of mass. We construct counterexamples to their conjecture in every characteristic. In this talk we shall discuss the technique of proof as well as the connection between escape of mass in continued fractions, Hecke trees, and number walls. This is part of ongoing works with Erez Nesharim and with Steven Robertson.