### 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

**2021**

**8**. Thursday, April 08, at 17:00 Moscow time (GMT+3).

**Slides of the lecture**

**Recording of the lecture**

**Speaker:** Lenny Fukshansky

**Title:** Representing integers by multilinear polynomials

**Abstract:**
Given a homogeneous multilinear polynomial F(x) in n variables with integer coefficients, we obtain some sufficient conditions for it to represent all the integers. Further, we derive effective results, establishing bounds on the size of a solution x to the equation F(x) = b, where b is any integer. For a special class of polynomials coming from determinants of rectangular matrices we are able to obtain necessary and sufficient conditions for such an effective representation problem. This result naturally connects to the problem of extending a collection of primitive vectors to a basis in a lattice, where we present counting estimates on the number of such extensions. The talk is based on joint works with A. Boettcher and with M. Forst.

**7**. Thursday, March 25, at 18:00 Moscow time (GMT+3).

**Slides of the lecture**

**Recording of the lecture**

**Speaker:** Dmitriy Bilyk

**Title:** Discrete minimizing measures

**Abstract:** In many natural examples, measures or point configurations minimizing various energy functionals tend to be uniformly distributed. However, a peculiar effect is observed for certain energies, especially with attractive-repulsive potentials: minimizing measures are not spread out over the domain, but rather turn out to be discrete (or, at least, supported on small sets). The collections of points that arise as discrete minimizers are often closely related to various objects and problems of discrete geometry: spherical designs, optimal codes, tight frames, equiangular lines, mutually unbiased bases, etc. We shall discuss this phenomenon and its applications, as well as the interplay between energy minimization and discrepancy theory.

**6**. Tuesday, March 23, at 15:00 Moscow time (GMT+3).

**Slides of the lecture**

**Recording of the lecture**

**Speaker:** Verónica Becher

**Title:** Nested perfect necklaces and normal numbers

**Abstract:** M. B. Levin used Sobol-Fauré low discrepancy sequences with Pascal triangle matrices modulo 2 to construct a real number x such that the first N terms of the sequence (2^n x mod 1)_{n≥1} have discrepancy O((log N)^2/N). This is the lowest discrepancy known for this kind of sequences. In this talk I will show that Levin’s construction can be characterized in terms of nested perfect necklaces, which are a variant of the classical de Bruijn sequences. Moreover, every real number x whose binary expansion is the concatenation of nested perfect necklaces of exponentially increasing order satisfies that the first N terms of (2^n x mod 1)_{n≥1} have discrepancy O((log N)^2/N). For the order being a power of 2, we know the exact number of nested perfect necklaces and an explicit method based on matrices to construct each of them.

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

**Slides of the lecture**

**Recording of the lecture**

**Speaker:** Alexander Gorodnik

**Title:** Can one count the «sound» of the space?

**Abstract:** We will explore some connections between Arithmetic (counting Diophantine solutions) and Analysis (spectral decomposition).

**4**. Tuesday, March 02, at 17:00 Moscow time (GMT+3).

**Recording of the lecture**

**Speaker:** Alex Kontorovich

**Title:** Diophantine Analysis on Groups

**Abstract:** We will describe some problems in geometry/arithmetic/group theory that can be attacked by methods from Diophantine analysis.

**3**. Tuesday, February 16, at 14:00 Moscow time (GMT+3).

**Slides of the lecture**

**Recording of the lecture**

**Speaker:** Anish Ghosh

**Title:** Diophantine approximation on varieties

**Abstract:** I will discuss some recent results in the theory of intrinsic Diophantine approximation on varieties with an emphasis on quantitative results.

**2**. Thursday, January 21, at 16:00 Moscow time (GMT+3).

**Slides of the lecture**

**Recording of the lecture**

**Speaker:** James Maynard

**Title:** Simultaneous small fractional parts of polynomials

**Abstract:** Given k real numbers \alpha_1,..,\alpha_k, how well can you simultaneously approximate these real numbers with squares? More specifically, how small can you make all the fractional parts \{n^2\alpha_1\}, … ,\{n^2\alpha_k\}$ using integers n<x? Classical work of Schmidt shows that there is an integer n<x such that all of these fractional parts are at most x^{-c/k^2} (for some constant c>0), uniformly for every choice of \alpha_i. I'll present some recent work which improves this bound to x^{-c/k}, which is optimal up to the value of the constant c>0 and similar results for more general polynomials instead of the squares. The improvement comes from a using fun blend of additive combinatorics and geometry of numbers.

**1**. Thursday, January 14, at 17:00 Moscow time (GMT+3).

**Slides of the lecture**

**Speaker:** Polina Vytnova

**Title:** Computing Hausdorff dimension of sets of continued fractions

**Abstract:** We will present a simple and practical approach to get rigorous bounds
on the Hausdorff dimension of limits sets of some one dimensional Markov iterated
function schemes which naturally arise in number theory, in particular in the study
of Markov and Lagrange spectra and in questions related to Zaremba conjecture

**2020**

**21**. Tuesday, December 15, at 14:00 Moscow time (GMT+3).

**Recording of the lecture**

**Speaker:** Carlos Matheus Santos

**Title:** On the intermediate portions of the Lagrange and Markov spectra

**Abstract:** The classical Lagrange and Markov spectra are closed subsets of the real line arising naturally in the study of Diophantine approximations of real numbers and certain indefinite binary quadratic forms. After the seminal works of A. Markov from 1879, the structure of these spectra were heavily investigated by many authors and, in particular, we know that these spectra begin with an explicit increasing sequence \sqrt{5} < \sqrt{8} < … converging to 3, and end with an explicit half-line [4.5278…, \infty).

On the other hand, several interesting questions about the intermediate portions of the Lagrange and Markov spectra are still open despite the progress obtained by Perron, Hall, Freiman, Flahive, … . In this talk, we will discuss some dynamical tools which might be relevant in piercing some of the mysteries about the structure of the intermediate regions in the classical spectra.

**20**. Thursday, December 10, at 18:00 Moscow time (GMT+3).

**Slides of the lecture**

**Speaker:** Tanguy Rivoal

**Title:** Algebraic values of E-functions

**Abstract:** I will present an algorithm that performs the following tasks: given an
E-function F (in the restricted sense) as input, it outputs the finite
list of algebraic points A such that F(A) is algebraic, together with
the list of the corresponding values F(A).

This is a joint work with Boris Adamczewski.

**19**. Tuesday, December 08, at 14:00 Moscow time (GMT+3).

**Speaker:** Dong Han Kim

**Title:** Intrinsic Diophantine Approximation of circles and spheres

**Abstract:** Let $S^1$ be the unit circle in $\mathbb{R}^2$ centered at the origin.
We study the intrinsic Diophantine approximation of $S_1$ and give a
complete description of the discrete part of the Lagrange spectrum.
We also consider the intrinsic Diophantine approximation of other
circles and spheres.
This talk is based on joint work with Byungchul Cha.

**18**. Tuesday, December 01, at 14:00 Moscow time (GMT+3).

**Recording of the lecture**

**Speaker:** Faustin Adiceam

**Title:** What does a sunflower look like in dimension 4?

**Abstract:** A more formal title for the talk could be: “the concept of bad approximability in geometric discrepancy”. Inspired by a most classical problem in phyllotaxis (the study of the arrangement of leaves on a plant stem), the goal will be to answer a question which has appeared in several places in the literature, namely: does there exist a spiral Delone set in any dimension? A typical example of such a point set visible in nature is the sunflower.
Here, a point set is said to be a spiral if it is obtained by suitable radial stretches of a spherical sequence. It is furthermore Delone if, roughly speaking, it is both well-spaced and dense in the Euclidean space.
We will see that the answer to the above question is related to several problems in the theory of the irregularity of distributions, in geometry (packing and covering of given objects, properties of Platonic solids…) and in Diophantine approximation. It is also closely related to the problem of equidistributing points on a sphere, a well-studied question to which many authors (including Arnol’d, Sarnak, Lubotsky, Phillips, Oh and Gorodnik) have contributed.
Time permitting, several open problems will conclude the talk. Joint work with Ioannis Tsokanos (University of Manchester).

**17**. Thursday, November 12, at 15:00 Moscow time (GMT+3).

**Slides of the lecture**

**Speaker:** Stéphane Fischler

**Title:** Linear independence of values of G-functions

**Abstract:** G-functions are a class of functions introduced by Siegel in 1929; they
include polylogarithms and $_{p+1} F_p$ hypergeometric series with
rational parameters. Given a transcendental $G$-function $F(z)$ and an
algebraic number $z_0$, it is in general a very difficult problem to
determine whether $F(z_0)$ is algebraic or transcendental, and even
whether it is rational or not. Seminal results due to Siegel, Galochkin,
Bombieri, Chudnovsky, André and others apply to $F(z_0)$ if $z_0$ is
sufficiently close to $0$, for instance proving its irrationality. On the
opposite, the point of view in this lecture (based upon a joint work with
Tanguy Rivoal) is to fix a non-zero algebraic point $z_0$, and to prove
irrationality results about $F(z_0)$ for some $G$-functions $F(z)$ in a
given family. This follows the approach of Ball-Rivoal, that yields the
irrationality of $\zeta(s)$ (i.e., the value at $z_0=1$ of the $s$-th
polylogarithm) for infinitely many odd integers $s$.

**16**. Tuesday, November 10, at 14:00 Moscow time (GMT+3).

**Recording of the lecture**

**Speaker:** Sam Chow

**Title:** Dyadic approximation in the Cantor set

**Abstract:** We investigate the approximation rate of a typical element of the Cantor set by dyadic rationals. This is a manifestation of Furstenberg's «times two times three» phenomenon, and is joint work with Demi Allen and Han Yu.

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

**Recording of the lecture**

**Speaker:** Alan Haynes

**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).

**Recording of the lecture**

**Speaker:** Florin Boca

**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.