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.
Date: 4 November 2025
Speaker: Reynold Fregoli
Title: Ergodic theorems for dilates of submanifolds in R^d actions
Abstract: I will discuss the validity of pointwise ergodic theorems for dilates of submanifolds in R^d-actions. In particular, I will present two recent results in this direction. The first, joint work with P. Bandi and D. Kleinbock, provides a positive result for continuous test functions in mixing R^d-actions. The second, joint work with J. Cheng and B. Guo, shows that if the regularity assumption on the test function is removed, a pointwise theorem may fail to hold.
Date: 18 November 2025
Speaker: Manuel Hauke
Title: Pseudo-random sequences, twin primes, and twisted diophantine approximation
Abstract: In this talk, I will speak about dynamics of $(a_n\alpha)_{n} \mod 1$ for integer sequences $(a_n)_n$ and fixed irrational rotations $\alpha$. The focus will be on the sequence of primes and other multiplicatively defined sequences, where gap statistics as well as twisted diophantine approximation will be considered. If time permits, I will outline the proof that includes a sieve coming from the twin prime counting problem, and establishing via random walks on Ostrowski digits an equidistribution result on diophantine Bohr sets mod d. This talk is partially based on https://arxiv.org/abs/2506.01736 and joint work with E. Kowalski https://arxiv.org/abs/2502.08335.
Date: 2 December 2025
Speaker: Zuo Lin
Title: Polynomial effective equidistribution for some higher dimensional unipotent subgroups
Abstract: Let G be a semisimple Lie group, Γ be a lattice in G and U be a unipotent subgroup of G. A celebrated theorem of Ratner says that for any x in G/Γ the orbit U.x is equidistributed in a periodic orbit of some subgroup U ≤ L ≤ G. Establishing a quantitative version of Ratner’s theorem has been long sought after. If U is a horospherical subgroup of G, the question is well-studied. If U is not a horospherical subgroup, this question is far less understood. Recently, Lindenstrauss, Mohammadi, Wang and Yang established a fully quantitative and effective equidistribution result for orbits of one-parameter (non-horospherical) unipotent groups in some cases. In this talk, we will discuss a recent equidistribution theorem for some unipotent subgroups in higher dimension. Our results in particular provide equidistribution theorems for orbits of the isometry group of a non-degenerate bilinear form on R^n in SL_n(R)/SL_n(Z).
Date: 9 December 2025
Speaker: Demi Allen
Title: Rectangular Shrinking Targets on Self-Similar Carpets
Abstract: Suppose $(X,d)$ is a metric space equipped with a Borel probability measure and suppose $(B_i)_{i \in \N}$ is a sequence of measurable sets in $X$. Suppose $T: X \to X$ is a measure preserving transformation and consider the set:
\[\cB = \{x \in X: T^n x \in B_n \text{ for infinitely many } n\in\N\}. \]
This is a shrinking target set. The terminology of “shrinking targets” was first introduced by Hill and Velani in 1995. Since then, shrinking target problems have received a great deal of interest, especially with regards to studying the measure-theoretic and dimension-theoretic properties of shrinking target sets. In this talk, I will discuss some recent work with Thomas Jordan (Bristol, UK) and Ben Ward (York, UK) where we establish the Hausdorff dimension of a shrinking target set where our “targets” (the $B_n$) are rectangles and $X$ is a self-similar carpet.
