### Содержание

# Спецкурсы и спецсеминары Николая Германовича Мощевитина

Написать Николаю Германовичу можно по почте: **moshchevitin@gmail.com** или в скайп: Nikolaus Moshchevitin

# Online seminar "Diophantine Analysis"

Local organizers: Nikolay Moshchevitin and Oleg German

The seminar is held in zoom.

If you are interested in participating please contact Nikolay Moshchevitin (**moshchevitin@gmail.com**).

## Lectures, new and old

**15**. Tuesday, October 06, at 16:00 Moscow time (GMT+3).

**Speaker:** Alan Haynes

**Recording of the lecture**

**Title:** Gap theorems for linear forms and for rotations on higher dimensional tori

**Abstract:** This talk is based on joint work with Jens Marklof, and with Roland Roeder. The three distance theorem states that, if x is any real number and N is any positive integer, the points x, 2x, … , Nx modulo 1 partition the unit interval into component intervals having at most 3 distinct lengths. We will present two higher dimensional analogues of this problem. In the first we consider points of the form mx+ny modulo 1, where x and y are real numbers and m and n are integers taken from an expanding set in the plane. This version of the problem was previously studied by Geelen and Simpson, Chevallier, Erdős, and many other people, and it is closely related to the Littlewood conjecture in Diophantine approximation. The second version of the problem is a straightforward generalization to rotations on higher dimensional tori which, surprisingly, has been largely overlooked in the literature. For the two dimensional torus, we are able to prove a five distance theorem, which is best possible. In higher dimensions we also have bounds, but establishing optimal bounds is an open problem.

**14**. Tuesday, September 29, at 14:00 Moscow time (GMT+3)..

**Speaker:** Florin Boca

**Recording of the lecture**

**Title:** Distribution of reduced quadratic irrationals (QIs) of even and of backward type

**Abstract:** Reduced QIs arising from the regular CF are closely related with the Pell
equation and with closed geodesics on the modular surface. By a classical result of
Pollicott, they are known to be equidistributed with respect to the Gauss probability
measure, when ordered by their corresponding closed primitive geodesics length.
This talk will consider the reduced quadratic irrationals arising from the even CF and
the backward CF, where similar (and effective) equidistribution results with respect to
the invariant (infinite!) invariant measures have been established in recent joint work
with M. Siskaki.

**13**. Tuesday, September 22, 2020, at 14:00 Moscow time (GMT+3).

**Slides of the lecture**

**Speaker:** Yitwah Cheung

**Title:** Mixing properties of the BCZ map

**Abstract:** The BCZ map was introduced by F. Boca, C. Cobeli and A. Zaharescu in their investigations of the statistical properties of the Farey sequences. This is a piecewise linear map of a certain right triangle whose connection to the horocycle flow on the modular surface was discovered by Athreya and myself, using which a newfound understanding of known results about Farey sequences was obtained, e.g. the distribution of gaps found by Hall in 1970 can be derived as the push-forward of Haar measure under the roof function of the BCZ map. In this talk I will report on joint work with Anthony Quas and Yiwei Zhang in our understanding of the mixing properties of the BCZ map. Specifically, I will explain how a miraculous property of the BCZ map, which we call infinitesimal self-similarity, allows us to prove that the BCZ map is weakly mixing.
The question of strong mixing remains open, correcting a claim I made earlier this year.
I will also describe a reformulation of the Riemann Hypothesis in terms of a BCZ cocycle.

**12**. Thursday, July 09, 2020, at 15:00 Moscow time (GMT+3).

**Speaker:** Reynold Fregoli

**Title:** Multiplicative badly approximable matrices and the Littlewood conjecture

**Abstract:** IS HERE

**11**. Thursday, June 25, 2020, at 15:00 Moscow time (GMT+3).

**Speaker:** Anthony Poëls.

**Title:** Rational approximation to real points on quadratic hypersurfaces.

**Abstract:** *Let Z be a quadratic hypersurface of R^n defined over Q containing points whose coordinates together with 1 are linearly independent over Q. In a joint work with Roy, we recently proved that, among these points, the largest exponent of uniform rational approximation is the inverse 1/rho_n of an explicit Pisot number rho_n < 2 depending only on n if the Witt index (over Q) of the quadratic form q defining Z is at most 1, and that it is equal to 1 otherwise. The proof for the upper bound 1/rho_n uses a recent transference inequality of Marnat and Moshchevitin. In the case n = 2, we recover results of Roy while for n > 2, this completes recent work of Kleinbock and Moshchevitin. We will explain the ideas behind the proofs and the constructions involved.*

**10**. Thursday, June 18, 2020, at 14:30 Moscow time (GMT+3).

**Speaker:** Nicolas Chevallier

**Title:** Minimal vectors in lattices over Gauss integers in C^2.

**Abstract:** The sequence of minimal vectors in a lattice can be seen as continued fraction expansion of the lattice. We will apply this idea to lattices over Gauss integers in C^2. Together with the sequences of minimal vectors, we will consider a submanifold in the space of unimodular lattices in C^2. Iterations of the first return map of the diagonal flow in this manifold are associated with the sequences of minimal vectors. This device provides a complex continued fraction expansion that should be related to A. Hurwitz complex continued fractions.

**9**. Thursday, June 11,2020, at 15:00 Moscow time (GMT+3).

**Speaker:** Jörg Thuswaldner.

**Title:** Multidimensional continued fractions and symbolic codings of toral translations

**Abstract:** The aim of this lecture is to find good symbolic codings for translations on the $d$-dimensional torus that enjoy the well-known and nice properties of Sturmian sequences (as for instance low complexity and good local discrepancy properties, i.e., bounded remainder sets of any scale). Inspired by the approach of G. Rauzy we construct such codings by the use of multidimensional continued fraction algorithms that are realized by sequences of substitutions.

**8**. Tuesday, June 02, 2020, at 15:00 Moscow time (GMT+3).

**Recording of the lecture**

**Speaker:** Erez Nesharim.

**Title:** The set of weighted badly approximable vectors is hyperplane absolute winning.

**Abstract:** We show that the set of badly approximable vectors with respect to any weight is hyperplane absolute winning. Our proof uses the quantitative nondivergence of unipotent flows in the space of lattices for absolutely friendly measures and the Cantor potential game. This is a recent work joint with Victor Beresnevich and Lei Yang.

**7**. Thursday, May 28, 2020, at 15:00 Moscow time (GMT+3).

**Recording of the lecture**

**Speaker:** Johannes Schleischitz

**Title:** Cartesian products, sumsets and Hausdorff dimension

**Abstract:** In 1962, Erdos proved that every real number is the sum of two Liouville
numbers. A direct consequence is that the Cartesian product L^2 of the set
of Liouville numbers with itself has Hausdorff dimension at least 1 (in
fact, equal), even though L has Hausdorff dimension 0. I will talk about
generalizations of this fact and point out other examples of sets that
naturally occur in Diophantine approximation whose Cartesian products have
unexpectedly large Hausdorff dimension.

**6**. Thursday, May 21, 2020, at 15:00 Moscow time (GMT+3).

**Recording of the lecture**

**Speaker:** Dmitri Kleinbock

**Title:** Geometry and dynamics of improvements to Dirichlet's Theorem in Diophantine approximation

**Abstract:** The set $\hat D^{m,n}$ of $m\times n$ matrices (systems of $m$ linear forms in $n$ variables) for which Dirichlet's theorem admits an improvement was originally studied by Davenport and Schmidt. They showed that the Lebesgue measure of $\hat D^{m,n}$ is zero, and that it contains the set of badly approximable matrices, hence has full Hausdorff dimension. A geometric approach to the notion of Dirichlet improvement identifies $\hat D^{m,n}$ with the set of lattices whose orbits stay away from the critical locus for the supremum norm. Based on that, I will present a generalized version of the Dirichlet improvement property and of theorems of Davenport and Schmidt. Joint work with Anurag Rao, and with Jinpeng An and Lifan Guan.

**5**. Thursday, May 07, 2020, at 14:30 Moscow time (GMT+3).

**Recording of the lecture**

**Speaker:** Victor Bereslevich

**Title:** Badly approximable points on curves are winning

**Abstract:** In this talk I will discuss a recent paper joint with Erez Nesharim and Lei Yang.

The main result of the paper shows that any set of weighted badly approximable points in R^n intersects any analytic non-degenerate curve in an absolute winning set. After a brief introduction including an account of previous results I will try to outline the main ideas of the proof.

**4**. Tuesday, May 05, 2020, at 14:00 Moscow time (GMT+3)

**Recording of the lecture**

**Speaker:** Dmitri Badziahin

**Title:** An improved bound in the problem of Wirsing

**Abstract:** For any real number x we define w_n^*(x) as the supremum of all positive real values w such that the inequality

|x - a| < H(a)^{-w-1}

has infinitely many solutions in algebraic real numbers of degree at most n. Here H(a) means the naive height of the minimal polynomial of a in Z[x] with coprime coefficients. In 1961, Wirsing asked whether the quantity w_n^*(x) as bounded from below by n for all transcendental x. Since then this problem remains mainly open. Wirsing himself only managed to establish the lower bound of the form w_n^*(x) \ge n/2+1 - o(1). Since then, the only improvements to this bound were in terms of O(1). I will talk about our resent work with Schleischitz where we managed to improve the bound by quantity O(n). More precisely, we show that w_n^*(x) > n/\sqrt{3}.

**3**. Tuesday, April 21, 2020, at 15:00 Moscow time (GMT+3).

**Recording of the lecture**

**Speaker:** Simon Kristensen

**Title:** Arithmetic properties of series of reciprocals of algebraic integers.

**Abstract:** Questions of irrationality of convergent series of reciprocals of integers is a fascinating one. Among other problems, the topic includes the question of irrationality of odd zeta-values. Of course, lower bounding the degree of such a number is a stronger and potentially harder problem.
I will introduce the problems studied within this field before proceeding with some recent research. In joint work with Simon Bruno Andersen, we provide a general growth criterion on a sequence of algebraic integers, which ensures that the degree of the series of reciprocals is transcendental or of degree at least D. Our result does not shed new light on the odd zeta-values, but it does extend results of Erdös as well as Hancl and Nair. Additionally, we will discuss analogous results for continued fractions.

**2**. Thursday April 16, 2020, at 15:00 Moscow time (GMT+3).

**Speaker:** Antoine Marnat.

**Title:** SINGULAR POINTS ON PRODUCT OF CERTAIN HOMOGENEOUS SPACES

**Abstract:** Recently, the study of singular vectors saw significant progress with, among others, work of S. Kadyrov, D. Kleinbock, E. Lindenstrauss and G.A. Margulis [3] and T. Das, L. Fishman, D. Simmons and M. Urbanski [2]. In joint work with J. An, L. Guan and R. Shi, we extend a result of Y Cheung [1] using techniques from the two previously mentioned papers to compute the Hausdorff dimension of singular and $\delta$-singular vectors on a product of unweighted homogeneous systems.

[1] Y. Cheung, Hausdorff dimension of the set of points on divergent trajectories of a homogeneousflow on a product space, Ergodic Theory and Dynamical Systems, 27(1), pp. 65–85 (2007) [2] T. Das, L. Fishman, D. Simmons and M.Urba ́nski,A variational principle in the parametricgeometry of numbers, ArXiV preprint 1901.06602. [3] S. Kadyrov, D. Kleinbock, E. Lindenstrauss and G.A. Margulis, Singular systems of linearforms and non-escape of mass in the space of lattices, J. Anal. Math.133(2017), 253–277.

**1**. Wednesday, April 08, 2020, at 13:00 Moscow time (GMT+3).

**Speaker:** Barak Weiss

**Title:** New bounds on the covering radius of a lattice

**Abstract:** We obtain new upper bounds on the minimal density of lattice coverings of R^n by dilates of a convex body K. We also obtain bounds on the probability (with respect to the natural Haar-Siegel measure on the space of lattices) that a randomly chosen lattice L satisfies L + K = R^n. As a step in the proof, we utilize and strengthen results on the discrete Kakeya problem. I will not assume any prior knowledge of lattice coverings. Joint with Or Ordentlich and Oded Regev.