Date: 30 September 2025
Speaker: Vasiliy Neckrasov
Title: Zero-one laws for uniform inhomogeneous Diophantine approximations
Abstract: In [Compositio Math. 155 (2019)] Kleinbock and Wadleigh proved a “zero-one law” for uniform Diophantine approximations to pairs (\Theta, \eta) of a matrix \Theta and vector \eta by using dynamics on the space of grids. We will show how the classical Diophantine transference principle provides an alternative approach to this problem and allows us to prove some generalizations. Namely, we will reduce the statement for pairs to the twisted (“fixed matrix”) case and show zero-one laws for twisted uniform approximations.
All the proofs are made in weighted case and, more generally, in the setup of approximations with arbitrary weight functions, which will also be discussed.
This talk is based on arXiv:2508.01912 and arXiv:2503.21180.
Date: 14 October 2025
Speaker: Chengyang Wu
Title: Simultaneously bounded and dense orbits for commuting Cartan actions
Abstract: With the goal to attack Uniform Littlewood’s Conjecture proposed in [BFK25], we introduced the concept of “fiberwise nondivergence” for the action of a cone inside the full diagonal subgroup of SL_3(R). Then it is proved in our paper that there exists a dense subset of SL_3(R)/SL_3(Z) in which each point has a fiberwise non-divergent orbit under a cone inside the full diagonal subgroup and an unbounded orbit under every diagonal flow. Our proof also presented the first instance of results concerning simultaneously bounded and dense orbits for commuting actions on noncompact spaces. This is a joint work with Dmitry Kleinbock.
Date: 21 October 2025
Speaker: Pratyush Sarkar
Title: Effective equidistribution of translates of tori in arithmetic homogeneous spaces and applications
Abstract: A celebrated theorem of Eskin–Mozes–Shah gives an asymptotic counting formula for the number of integral (n x n)-matrices with a prescribed irreducible (over the integers/rationals) integral characteristic polynomial. We obtain a power saving error term for the counting problem for (3 x 3)-matrices. We do this by using the connection to homogeneous dynamics and proving effective equidistribution of translates of tori in SL_3(R)/SL_3(Z). A key tool is that the limiting Lie algebra corresponding to the translates of tori is a certain nilpotent Lie algebra. This allows us to use the recent breakthrough work of Lindenstrauss–Mohammadi–Wang–Yang on effective versions of Shah’s/Ratner’s theorems. We actually study the phenomenon more generally for any semisimple Lie group which we may discuss if time permits.
Date: 28 October 2025
Speaker: Suxuan Chen
Title: The Hausdorff dimension of the intersection of \psi-well approximable numbers and self-similar sets
Abstract: Let \psi be a monotonically non-increasing function from N to R, and let \psi_v be defined by \psi_v(q)=1/q^v. Here, we consider self-similar sets whose iterated function systems satisfy the open set condition. For functions \psi that do not decrease too rapidly, we give a conjecturally sharp upper bound on the Hausdorff dimension of the intersection of \psi-well approximable numbers and such self-similar sets. When \psi=\psi_v for some v greater than 1 and sufficiently close to 1, we give a lower bound for this Hausdorff dimension, which asymptotically matches the upper bound as v approaches 1. In particular, we show that the set of very well approximable numbers has full Hausdorff dimension within self-similar sets.
