This online seminar is devoted to recent progress in Diophantine approximation, homogeneous dynamics and related topics. Every lecture will consist of two parts: the first part for a general introduction and the second part for explaining proofs in detail.

Talks will be given on zoom. Please send an email to one of the organizers to get the access.

If you would like to give a talk, please send the title, abstract and related papers (if available) of your proposed talk to one of the organizers by email.

Organizers: Victor Beresnevich, Erez Nesharim, Lei Yang

All Talks

### Title: An inhomogeneous Khintchine-Groshev Theorem without monotonicity

### Title: Higher moment formulas for Siegel transforms and applications to limit distributions of functions of counting lattice points

### Title: Invariants of Diophantine Approximation

### Title: Shrinking target and horocycle equidistribution

Speaker: Demi Allen (University of Warwick)

Date: 17 December, 2021 14:30 UTC

Abstract: The classical (inhomogeneous) Khintchine-Groshev Theorem tells us that for a monotonic approximating function $\psi: \mathbb{N} \to [0,\infty)$ the Lebesgue measure of the set of (inhomogeneously) $\psi$-well-approximable points in $\mathbb{R}^{nm}$ is zero or full depending on, respectively, the convergence or divergence of $\sum_{q=1}^{\infty}{q^{n-1}\psi(q)^m}$. In the homogeneous case, it is now known that the monotonicity condition on $\psi$ can be removed whenever $nm>1$, and cannot be removed when $nm=1$. In this talk I will discuss recent work with Felipe A. Ramírez (Wesleyan, US) in which we show that the inhomogeneous Khintchine-Groshev Theorem is true without the monotonicity assumption on $\psi$ whenever $nm>2$. This result brings the inhomogeneous theory almost in line with the completed homogeneous theory. I will survey previous results towards removing monotonicity from the homogeneous and inhomogeneous Khintchine-Groshev Theorem before discussing the main ideas behind the proof our recent result.

Full InformationSpeaker: Jiyoung Han(Tata Institute)

Date: 3 December, 2021 14:30 UTC

Abstract: The Siegel transform is one of the main tools when we consider problems related to counting lattice points using homogeneous dynamics. It is revealed by many mathematicians that Siegel’s integral formula and Rogers’ second moment formula are very useful to solve various quantitative and effective variants of classical problems in the geometry of numbers, such as the Gauss circle problem (generalized to convex sets) and Oppenheim conjecture. Furthermore, Rogers’ higher moment formulas, together with the method of moments, give us information about limit distributions related to these problems. In this talk, we revisit Rogers’ higher moment formulas with a new approach, and introduce higher moment formulas for Siegel transforms on the space of affine unimodular lattices and the space of unimodular lattices with a congruence condition. Using these formulas, we obtain the results of limit distributions, which are generalizations of the work of Rogers (1956), Södergren (2011), and Strömbergsson... Full Information

Speaker: Yitwah Cheung (Tsinghua University)

Date: 19 November, 2021 14:30 UTC

Abstract: There is a natural generalization of the concept of convergents of the continued fraction to higher dimensions that does not involve any specific choice of norm. In this talk, I will motivate this concept from several different angles, within the framework of staircases, which is a rectilinear version of Kleinian sails. I will describe some results about dual convergents and illustrate the method of our approach towards constructing slowly unbounded A-orbits by sketching the proof of dichotomy of Hausdorff dimension phenomenon obtained in joint work with P. Hubert and H. Masur.

Full InformationSpeaker: Jimmy Tseng (Exeter)

Date: 5 November, 2021 14:30 UTC

Abstract: Consider a shrinking neighborhood of a cusp of the unit tangent bundle of a noncompact hyperbolic surface of finite area. We discuss how a closed horocycle whose length goes to infinity can become equidistributed on this shrinking neighborhood, giving a sharp criterion in a natural case. This setup is closely related to number theory, and, as an example, our method yields a number-theoretic identity.

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