researchseminars.org

Osama Khalil (The University of Utah)
2020-06-03 14:00 - 15:30 UTC | https://stable.researchseminars.org/talk/DAHD-webinar/1/
Title: Random walks, spectral gaps, and Khintchine's theorem on fractals (https://stable.researchseminars.org/talk/DAHD-webinar/1/)
by Osama Khalil (The University of Utah) as part of Webinar on Diophantine approximation and homogeneous dynamics

Abstract
In 1984, Mahler asked how well typical points on Cantor’s set can be approximated by rational numbers. His question fits within a program, set out by himself in the 1930s, attempting to determine conditions under which subsets of $\mathbb{R}^d$ inherit the Diophantine properties of the ambient space. Since the approximability of typical points in Euclidean space by rational points is governed by Khintchine’s classical theorem, the ultimate form of Mahler’s question asks whether an analogous zero-one law holds for fractal measures. Significant progress has been achieved in recent years, albeit, almost all known results have been of “convergence type”.
In this talk, we will discuss the first instances where a complete analogue of Khinchine’s theorem for fractal measures is obtained. The class of fractals for which our results hold includes those generated by rational similarities of $\mathbb{R}^d$ and having sufficiently small Hausdorff co-dimension. The main new ingredient is an effective equidistribution theorem for certain fractal measures on the space of unimodular lattices. The latter is established via a new technique involving the construction of $S$-arithmetic Markov operators possessing a spectral gap and encoding the arithmetic structure of the maps generating the fractal. This is joint work in progress with Manuel Luethi.

Barak Weiss (Tel Aviv University)
2020-06-10 12:30 - 14:00 | https://stable.researchseminars.org/talk/DAHD-webinar/2/
Title: New bounds on the covering density of a lattice (https://stable.researchseminars.org/talk/DAHD-webinar/2/)
by Barak Weiss (Tel Aviv University) as part of Webinar on Diophantine approximation and homogeneous dynamics

Abstract
We obtain new upper bounds on the minimal density of lattice coverings of $\mathbb{R}^n$ 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 $L+K=\mathbb{R}^n$. Joint work with Or Ordentlich and Oded Regev.

James Maynard (University of Oxford)
2020-06-17 13:00 - 15:00 | https://stable.researchseminars.org/talk/DAHD-webinar/3/
Title: On the Duffin-Schaeffer Conjecture (https://stable.researchseminars.org/talk/DAHD-webinar/3/)
by James Maynard (University of Oxford) as part of Webinar on Diophantine approximation and homogeneous dynamics

Abstract
Almost 80 years ago Duffin and Schaeffer conjectured a beautiful strengthening of Khinchin's classical result: Given a sequence of possible forms of rational approximation, either almost all reals can be approximated in this manner or almost none can be, and there is a simple calculation to tell which case we are in.
I'll talk about recent work with D. Koukoulopoulos which establishes this conjecture. This relies on a blend of different techniques, recasting the problem as a structural question in additive combinatorics, and then approaching this via studying a particular family of graphs to reduce it to a problem in analytic number theory.

Lei Yang (Sichuan University)
2020-07-01 13:30 - 15:30 | https://stable.researchseminars.org/talk/DAHD-webinar/4/
Title: Winning property of badly approximable points on curves (https://stable.researchseminars.org/talk/DAHD-webinar/4/)
by Lei Yang (Sichuan University) as part of Webinar on Diophantine approximation and homogeneous dynamics

Abstract
We will prove that badly approximable points (no matter weighted or unweighted) on any analytic non-degenerate curve in $\mathbb{R}^n$ is an absolute winning set. This confirms a key conjecture in the area stated by Badziahin and Velani (2014) which represents a far-reaching generalisation of Davenport's problem from the 1960s. Amongst various consequences of our main result is a solution to Bugeaud's problem on real numbers badly approximable by algebraic numbers of arbitrary degree. This work is joint with Victor Beresnevich and Erez Nesharim.

Nikolay Moshchevitin (Moscow State University)
2020-07-08 13:30 - 15:30 | https://stable.researchseminars.org/talk/DAHD-webinar/5/
Title: Singular vectors: from Khintchine to nowadays (https://stable.researchseminars.org/talk/DAHD-webinar/5/)
by Nikolay Moshchevitin (Moscow State University) as part of Webinar on Diophantine approximation and homogeneous dynamics

Abstract
Many concepts in Diophantine Approximation have their origin in the
famous paper "Über eine Klasse linearer diophantischer Approximationen" by A. Khintchine (1926). The results of this paper were rediscovered many times by different mathematicians. In particular, Khintchine was the first who observed the phenomenon of singularity in higher-dimensional Diophantine Approximation. In our lecture we discuss several problems related to singular vectors and best approximation (minimal points) as well as some related topics dealing with Diophantine exponents and approximation on algebraic and analytic surfaces which were considered recently in author's joint papers with D. Kleinbock and B. Weiss. Also we suppose to discuss some related open problems.

Baowei Wang (Huazhong University of Sci. & Tech.)
2020-07-22 13:30 - 14:30 | https://stable.researchseminars.org/talk/DAHD-webinar/6/
Title: Mass transference principle from rectangles to rectangles in Diophantine approximation (https://stable.researchseminars.org/talk/DAHD-webinar/6/)
by Baowei Wang (Huazhong University of Sci. & Tech.) as part of Webinar on Diophantine approximation and homogeneous dynamics

Abstract
As is well known, Dirichlet's theorem and Minkowski's theorem are two fundamental results in Diophantine approximation. One says that all points in R^d will fall into infinitely many balls centered at rationals with specific radius; while the other says that all points will fall into infinitely many rectangles centered at rationals with specific sidelengths. This motives a further study on the metric theory of limsup sets defined by a sequence of balls or rectangles. Since the landmark works of Beresnevich & Velani (2006) and Beresnevich, Dickinson & Velani (2006) where the mass transference principle was found, the metric theory for limsup sets defined by a sequence balls or isotropic thicken of general sets has been sufficiently well established. While, the metric theory for limsup sets defined by a sequence of rectangles are not as rich as the ball case. In this talk, I will talk about some progess on the metric theory of the latter case by modifying the settings in the above mentioned impressing works.

This talk will only take one hour.

Pengyu Yang (ETH Zurich)
2020-07-29 13:30 - 15:30 | https://stable.researchseminars.org/talk/DAHD-webinar/7/
Title: Dirichlet Improvability, Equidistribution, and Grassmannians (https://stable.researchseminars.org/talk/DAHD-webinar/7/)
by Pengyu Yang (ETH Zurich) as part of Webinar on Diophantine approximation and homogeneous dynamics

Abstract
As a natural generalisation of Dirichlet's approximation theorem on real numbers, Dirichlet's approximation theorem on $m\times n$ real matrices tells us the following: given $m$ real linear forms in $n$ variables, we can find an integral vector such that the evaluations of all the linear forms at this integral vector are simultaneous small. In the 1960s Davenport and Schmidt showed that Dirichlet’s theorem is non-improvable for almost all matrices, and they asked if the analogous result holds for a submanifold of the space of $m\times n$ matrices. This problem is related to an equidistribution problem in the space of unimodular lattices in $\mathbb{R}^n$. In this talk I will present some recent progress on this problem, and I will explain its connections to the geometry of Grassmannian manifolds.

Anurag Rao (Brandeis University)
2020-08-05 13:00 - 14:00 | https://stable.researchseminars.org/talk/DAHD-webinar/8/
Title: Some problems in uniform Diophantine approximation (https://stable.researchseminars.org/talk/DAHD-webinar/8/)
by Anurag Rao (Brandeis University) as part of Webinar on Diophantine approximation and homogeneous dynamics

Abstract
We study a norm sensitive Diophantine approximation problem arising from the work of Davenport and Schmidt on the improvement of Dirichlet's theorem. Its supremum norm case was recently considered by the Kleinbock and Wadleigh, and here we extend the set-up by replacing the supremum norm with an arbitrary norm. This gives rise to a class of shrinking target problems for one-parameter diagonal flows on the space of lattices, with the targets being neighborhoods of the critical locus of a suitably scaled norm ball. We use methods from geometry of numbers and dynamics to generalize a result due to Andersen and Duke on measure zero and uncountability of the set of numbers for which Minkowski approximation theorem can be improved. The choice of the Euclidean norm on $\mathbb{R}^2$ corresponds to studying geodesics on a hyperbolic surface which visit a decreasing family of balls. An application of a dynamical Borel-Cantelli lemma of Maucourant produces a zero-one law for improvement of Dirichlet's theorem in Euclidean norm. Based on joint works with Dmitry Kleinbock and Srinivasan Sathiamurthy.

The starting time is earlier than usual. The talk will only take one hour.

Anish Ghosh (Tata Institute of Fundamental Research)
2020-08-12 13:30 - 15:30 | https://stable.researchseminars.org/talk/DAHD-webinar/9/
Title: Diophantine approximations, large intersections and geodesics in negative curvature (https://stable.researchseminars.org/talk/DAHD-webinar/9/)
by Anish Ghosh (Tata Institute of Fundamental Research) as part of Webinar on Diophantine approximation and homogeneous dynamics

Abstract
I will discuss new results on the `shrinking target problem' including a logarithm law for approximation by geodesics in negatively curved manifolds and Hausdorff dimension estimates for finer spiraling phenomena of geodesics. I will also discuss the large intersection property of Falconer in the context of negative curvature and some applications to Diophantine approximation and to hyperbolic geometry. This is joint work with Debanjan Nandi.

Denis Koleda (Institute of mathematics, Minsk)
2020-08-19 13:30 - 15:30 | https://stable.researchseminars.org/talk/DAHD-webinar/10/
Title: The distribution of conjugate algebraic numbers: a random polynomial approach (https://stable.researchseminars.org/talk/DAHD-webinar/10/)
by Denis Koleda (Institute of mathematics, Minsk) as part of Webinar on Diophantine approximation and homogeneous dynamics

Abstract
In the talk we consider the spatial distribution of points that have algebraic (Galois) conjugate coordinates of fixed degree and bounded height. We give an asymptotic formula for counting such points in a wide class of regions of Euclidean space (as the parameter that bounds heights grows to infinity). We explain connection of this formula to random polynomials with i.i.d. coefficients. We also discuss some corollaries and applications of the formula. The talk is based on a joint work with F. Götze and D. Zaporozhets.

Kaisa Matomäki (University of Turku)
2020-08-26 13:30 - 15:30 | https://stable.researchseminars.org/talk/DAHD-webinar/11/
Title: Higher order uniformity of the Liouville function (https://stable.researchseminars.org/talk/DAHD-webinar/11/)
by Kaisa Matomäki (University of Turku) as part of Webinar on Diophantine approximation and homogeneous dynamics

Abstract
The Liouville function takes a value +1 or -1 at a natural number $n$ depending on whether $n$ has an even or an odd number of prime factors. The Liouville function is believed to behave more or less randomly. In particular a famous conjecture of Sarnak says that the Liouville function does not correlate with any sequence of "low complexity" whereas a longstanding conjecture of Chowla says that the Liouville function has negligible correlations with its own shifts.
I will discuss conjectures of Sarnak and Chowla and my very recent work with Radziwiłł, Tao, Teräväinen, and Ziegler, where we show that, in almost all intervals of length $X^\varepsilon$, the Liouville function does not correlate with polynomial phases or more generally with nilsequences. I will also discuss applications to superpolynomial word complexity for the Liouville sequence and to a new averaged version of Chowla's conjecture.

Damien Roy (University of Ottawa)
2020-09-23 13:30 - 15:30 | https://stable.researchseminars.org/talk/DAHD-webinar/12/
Title: Simultaneous rational approximation to exponentials of algebraic numbers (https://stable.researchseminars.org/talk/DAHD-webinar/12/)
by Damien Roy (University of Ottawa) as part of Webinar on Diophantine approximation and homogeneous dynamics

Abstract
The theorem of Lindemann-Weierstrass asserts that the exponentials of distinct algebraic numbers are linearly independent over the field of rational numbers. The proof uses a construction of simultaneous rational approximations to such exponentials values, which goes back to Hermite. In this talk, we show that, from an adelic perspective, these approximations are essentially best possible. This point of view partly explains the nature of the algebraic numbers whose exponentials have a structured continuous fraction expansion. We also propose few specific conjectures regarding simultaneous approximations to such values in adèle rings.

The proof of our main result requires a separate analysis for each place of the associated number field. For the Archimedean places, it relies on the structure of the graph drawn in the complex plane by the paths of fastest descent for the norm of a general univariate complex polynomial starting from the roots of its derivative and ending in the roots of the polynomial. It happens that this graph is a tree and that the lengths of those paths can be estimated from above in terms of the degree of the given polynomial and the diameter of its set of zeros. We will mention some instances where these upper bounds can be greatly improved and state as an open problem whether or not such improvements hold in general.

Sam Chow (University of Warwick)
2020-09-16 13:30 - 15:30 | https://stable.researchseminars.org/talk/DAHD-webinar/13/
Title: A fully-inhomogeneous version of Gallagher's theorem (https://stable.researchseminars.org/talk/DAHD-webinar/13/)
by Sam Chow (University of Warwick) as part of Webinar on Diophantine approximation and homogeneous dynamics

Abstract
Gallagher's theorem describes the multiplicative diophantine
approximation rate of a typical vector. We establish a fully-inhomogeneous
version of Gallagher's theorem, a diophantine fibre refinement, and a
sharp and unexpected threshold for Liouville fibres. Along the way, we
prove an inhomogeneous version of the Duffin--Schaeffer conjecture for a
class of non-monotonic approximation functions. Joint with Niclas Technau.

Ralf Spatzier (University of Michigan)
2020-09-30 13:30 - 15:30 | https://stable.researchseminars.org/talk/DAHD-webinar/14/
Title: Hyperbolic Actions of Higher Rank Abelian Groups (https://stable.researchseminars.org/talk/DAHD-webinar/14/)
by Ralf Spatzier (University of Michigan) as part of Webinar on Diophantine approximation and homogeneous dynamics

Abstract
We study transitive $\mathbb{R} ^k \times \mathbb{Z}^\ell$ actions on arbitrary compact manifolds with a projectively dense set of Anosov elements and 1-dimensional coarse Lyapunov foliations. Such actions are called totally Cartan actions. We completely classify such actions as built from low-dimensional Anosov flows and diffeomorphisms and affine actions, verifying the Katok-Spatzier conjecture for this class. This is achieved by introducing a new tool, the action of a dynamically defined topological group describing paths in coarse Lyapunov foliations, and understanding its generators and relations. We obtain applications to the Zimmer program. This talk is based on joint work with Kurt Vinhage.

Mishel Skenderi (Brandeis University)
2020-10-14 13:30 - 15:30 | https://stable.researchseminars.org/talk/DAHD-webinar/15/
Title: Small values at integer points of generic subhomogeneous functions (https://stable.researchseminars.org/talk/DAHD-webinar/15/)
by Mishel Skenderi (Brandeis University) as part of Webinar on Diophantine approximation and homogeneous dynamics

Abstract
This talk will be based on joint work with Dmitry Kleinbock that has been motivated by several recent papers (among them, those of Athreya-Margulis, Bourgain, Ghosh-Gorodnik-Nevo, Kelmer-Yu). Given a certain sort of group $G$ and certain sorts of functions $f: \mathbb{R}^n \to \mathbb{R}$ and $\psi : \mathbb{R}^n \to \mathbb{R}_{>0}$, we obtain necessary and sufficient conditions so that for Haar-almost every $g \in G$, there exist infinitely many (respectively, finitely many) $v \in \mathbb{Z}^n$ for which $|(f \circ g)(v)| \leq \psi(\|v\|)$, where $\|\cdot\|$ is an arbitrary norm on $\mathbb{R}^n$. We also give a sufficient condition in the setting of uniform approximation. As a consequence of our methods, we obtain generalizations to the case of vector-valued (simultaneous) approximation with no additional effort. In our work, we use probabilistic results in the geometry of numbers that go back several decades to the work of Siegel, Rogers, and W. Schmidt; these results have recently found new life thanks to a 2009 paper of Athreya-Margulis.

Jialun Li (University of Zurich)
2020-10-21 13:30 - 15:30 | https://stable.researchseminars.org/talk/DAHD-webinar/16/
Title: Decay of Fourier transforms of fractal measures (https://stable.researchseminars.org/talk/DAHD-webinar/16/)
by Jialun Li (University of Zurich) as part of Webinar on Diophantine approximation and homogeneous dynamics

Abstract
We will talk about some of the recent works on estimating the decay of Fourier transforms of fractal measures, such as self-similar measures and Furstenberg measures. The proof is based on renewal theorems for stopping times of random walks on $\mathbb{R}$.

Alessandro Pezzoni (University of York)
2020-10-28 14:30 - 16:30 | https://stable.researchseminars.org/talk/DAHD-webinar/17/
Title: Simultaneous Diophantine approximation on manifolds by algebraic numbers (https://stable.researchseminars.org/talk/DAHD-webinar/17/)
by Alessandro Pezzoni (University of York) as part of Webinar on Diophantine approximation and homogeneous dynamics

Abstract
Simultaneous Diophantine approximation on manifolds is notoriously
complicated, since it requires to take into account the arithmetic
properties of a manifold, as well as the analytic ones. In this talk
we will make some progress towards a metric theory of approximation on
manifolds by algebraic numbers with algebraic conjugate coordinates,
generalising a conjecture of Sprindžuk's.

Antoine Marnat (TU Graz)
2020-11-04 14:30 - 16:30 | https://stable.researchseminars.org/talk/DAHD-webinar/18/
Title: Dirichlet is not just Bad and Sing (https://stable.researchseminars.org/talk/DAHD-webinar/18/)
by Antoine Marnat (TU Graz) as part of Webinar on Diophantine approximation and homogeneous dynamics

Abstract
It is well known that in dimension one, the set of Dirichlet
improvable real numbers consists precisely of badly approximable and
singular numbers. We show that in higher dimensions this disjoint union
is not the full set of Dirichlet improvable vectors: we prove that there
exist uncountably many Dirichlet improvable vectors that are neither
badly approximable nor singular. We construct these numbers using the
parametric geometry of numbers. Furthermore, by doing so we can choose
the exponent of Diophantine approximation by a rational subspace of
dimension exactly $d$, for any d between $0$ and $n-1$.

Alan Haynes (University of Houston)
2020-11-11 13:30 - 15:30 | https://stable.researchseminars.org/talk/DAHD-webinar/19/
Title: Higher dimensional gap theorems in Diophantine approximation (https://stable.researchseminars.org/talk/DAHD-webinar/19/)
by Alan Haynes (University of Houston) as part of Webinar on Diophantine approximation and homogeneous dynamics

Abstract
The three distance theorem states that, if $x$ is any real number and $N$ is any positive integer, the points $x, 2x, … , Nx \mod 1$ partition the unit interval into component intervals having at most $3$ distinct lengths. There are many higher dimensional analogues of this theorem, and in this talk we will discuss two of them. In the first we consider points of the form $mx+ny \mod 1$, 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. The first hour of this talk will be expository, and the second half will focus on proofs. The new results presented in this talk are joint work with Jens Marklof and with Roland Roeder.

Michael Bersudsky (IIT Technion)
2020-11-18 14:30 - 16:30 | https://stable.researchseminars.org/talk/DAHD-webinar/20/
Title: On the image in the torus of sparse points on expanding analytic curves (https://stable.researchseminars.org/talk/DAHD-webinar/20/)
by Michael Bersudsky (IIT Technion) as part of Webinar on Diophantine approximation and homogeneous dynamics

Abstract
It is known that the projection to the $2$-torus of the normalised parameter measure on a circle of radius $R$ in the plane becomes uniformly distributed as $R$ grows to infinity. I will discuss the following natural discrete analogue for this problem. Starting from an angle and a sequence of radii $\{R_n\}$ which diverges to infinity, I will consider the projection to the 2-torus of the $n$'th roots of unity rotated by this angle and dilated by a factor of $R_n$. The interesting regime in this problem is when $R_n$ is much larger than $n$ so that the dilated roots of unity appear sparsely on the dilated circle. I will discuss 3 types of results:

1. Validity of equidistribution for all angles when the sparsity is polynomial.

2. Failure of equidistribution for some super polynomial dilations.

3. Equidistribution for almost all angles for arbitrary dilations.

I will then pass to discuss more general results on the projection to the $d$-torus of dilations of varying analytic curves in $d$-space.

Han Yu (University of Cambridge)
2020-11-25 14:30 - 16:30 | https://stable.researchseminars.org/talk/DAHD-webinar/21/
Title: Rational points near self-similar sets (https://stable.researchseminars.org/talk/DAHD-webinar/21/)
by Han Yu (University of Cambridge) as part of Webinar on Diophantine approximation and homogeneous dynamics

Abstract
We show that the rational points are quite well
'equidistributed' near the middle 15th Cantor set $K$. As a consequence,
it is possible to show that the set of well-approximable numbers has
full Hausdorff dimension inside $K$. This answers a question of
Levesley-Salp-Velani for $K$. In fact, it is possible to prove a
slightly stronger result which partially answers a question of
Bugeaud-Durand. The results also hold for some self-similar sets other
than $K$. We will provide a sufficient condition and some other
examples. We suspect that the above results hold for all self-similar
sets with Hausdorff dimension bigger than $1/2$ and with the open set
condition. We will see some heuristics in the talk.

Shreyasi Datta (University of Michigan)
2020-12-02 14:30 - 16:30 | https://stable.researchseminars.org/talk/DAHD-webinar/22/
Title: Recent progress in p-adic Diophantine approximation (https://stable.researchseminars.org/talk/DAHD-webinar/22/)
by Shreyasi Datta (University of Michigan) as part of Webinar on Diophantine approximation and homogeneous dynamics

Abstract
Studying the $p$-adic analogue of Mahler's conjecture was initiated by Sprind zuk in 1969. Subsequently, there were several partial results culminating in the work of Kleinbock and Tomanov, where the $S$-adic case of the Baker-Sprindzuk conjectures were settled in full generality. We provide a complete $p$-adic analogue of the results of D. Kleinbock on Diophantine exponents of affine subspaces. This answers a conjecture of Kleinbock and Tomanov. Recently, we proved $S$-arithmetic inhomogeneous Khintchine type theorems on analytic nondegenerate manifolds. For $S$ consisting of more than one valuation, the divergence results are new even in the homogeneous setting. This aformentioned result answers questions posed by Badziahin, Beresnevich and Velani and also it generalizes the work of Golsefidy and Mohammadi. In the first half of the talk, I will go over these results and in the seocnd half, I will try to concentrate on some of the technical details of the proofs. The new results presented in this talk are joint work with Anish Ghosh.

Shahriar Mirzadeh (Michigan State University)
2020-12-09 14:30 - 16:30 | https://stable.researchseminars.org/talk/DAHD-webinar/23/
Title: Upper bound for the Hausdorff dimension of exceptional orbits in homogeneous spaces (https://stable.researchseminars.org/talk/DAHD-webinar/23/)
by Shahriar Mirzadeh (Michigan State University) as part of Webinar on Diophantine approximation and homogeneous dynamics

Abstract
Consider the set of points in a homogeneous space $G/\Gamma$
whose $g_t$-orbit misses a fixed open set. It has measure zero
if the flow is ergodic. It has been conjectured that this set has
Hausdorff dimension strictly smaller than the dimension of whole space. This
conjecture is proved when $G/\Gamma$ is compact or when has real rank. In
this talk we will prove the conjecture for probably the most important
example of the higher rank case namely: $\SL(m+n, \R)/\SL(m+n, \Z)$ and $g_t = \mathrm{diag}\{e^{t/m}, \dots , e^{t/m}, e^{-t/n}, \dots, e^{-t/n}\}$. This
homogeneous space has many applications in Diophantine
approximation that will be discussed in the talk if time permits. This
project is joint work with Dmitry Kleinbock.

Svetlana Jitomirskaya (UC Irvine)
2020-12-16 16:00 - 18:00 | https://stable.researchseminars.org/talk/DAHD-webinar/24/
Title: Inhomogeneous Diophantine approximation in the coprime setting (https://stable.researchseminars.org/talk/DAHD-webinar/24/)
by Svetlana Jitomirskaya (UC Irvine) as part of Webinar on Diophantine approximation and homogeneous dynamics

Abstract
Given $n\in
$ and $x\in\R$, let
$$||nx||^\prime=\min\{|nx-m|:m\in\Z, gcd(n,m)=1\}.$$
Two conjectures in the coprime inhomogeneous Diophantine approximation stated, by analogy with the classical Diophantine approximation, that for any irrational number $\alpha$ and almost every $\gamma\in \R$,
$$\liminf_{n\to \infty}n||\gamma -n\alpha||^{\prime}=0,$$
and that there exists $C$ such that for all $\gamma\in \R$,

$$\liminf_{n\to \infty}n||\gamma -n\alpha||^{\prime} < C.$$

We will present our joint work with W. Liu that proves one of those and disproves the other.

Manos Zafeiropoulos (TU Graz)
2021-02-10 14:30 - 16:30 | https://stable.researchseminars.org/talk/DAHD-webinar/25/
Title: Inhomogeneous Diophantine Approximation on $M_0$ Sets with restricted denominators (https://stable.researchseminars.org/talk/DAHD-webinar/25/)
by Manos Zafeiropoulos (TU Graz) as part of Webinar on Diophantine approximation and homogeneous dynamics

Abstract
Let $\mu$ be a probability measure with $\widehat{\mu}(t)\ll (\log |t|)^{-A}$ for some $A>0$, supported on a set $F\subseteq [0,1]$. Let $\mathcal{A}=(q_n)_{n=1}^{\infty}$ be an increasing
sequence of integers. We establish a quantitative inhomogeneous Khintchine-type theorem in which the points of interest lie in $F$ and the "denominators" of the approximants belong to $\mathcal{A}$ in the following cases:
(i) $(q_n)_{n=1}^{\infty}$ is lacunary and $A>2$.
(ii)The prime divisors of $(q_n)_{n=1}^{\infty}$ are restricted in a set of $k$ prime numbers and $A>2k$.

Boris Adamczewski (Claude Bernard University Lyon 1)
2021-02-17 14:30 - 16:30 | https://stable.researchseminars.org/talk/DAHD-webinar/26/
Title: Furstenberg's conjecture, Mahler's method, and finite automata (https://stable.researchseminars.org/talk/DAHD-webinar/26/)
by Boris Adamczewski (Claude Bernard University Lyon 1) as part of Webinar on Diophantine approximation and homogeneous dynamics

Abstract
It is commonly expected that expansions of numbers in multiplicatively independent bases, such as 2 and 10, should have no common structure. However, it seems extraordinarily difficult to confirm this naive heuristic principle in some way or another. In the late 1960s, Furstenberg suggested a series of conjectures, which became famous and aim to capture this heuristic. The work I will discuss in this talk is motivated by one of these conjectures. Despite recent remarkable progress by Shmerkin and Wu, it remains totally out of reach of the current methods. While Furstenberg’s conjectures take place in a dynamical setting, I will use instead the language of automata theory to formulate some related conjectures that formalize and express in a different way the same general heuristic. I will explain how the latter follow from some recent advances in Mahler's method; a method in transcendental number theory initiated by Mahler in the end of the 1920s. This a joint work with Colin Faverjon.

Dmitry Badziahin (University of Sydney)
2021-02-24 10:00 - 12:00 | https://stable.researchseminars.org/talk/DAHD-webinar/27/
Title: Diophantine approximation on the Veronese curve (https://stable.researchseminars.org/talk/DAHD-webinar/27/)
by Dmitry Badziahin (University of Sydney) as part of Webinar on Diophantine approximation and homogeneous dynamics

Abstract
In the talk we discuss the set $S_n(\lambda)$ of simultaneously $\lambda$-well approximable points in $\mathbb{R}^n$. These are the points $x$ such that the inequality
$$\| x - p/q\|_\infty < q^{-\lambda - \epsilon}$$
has infinitely many solutions in rational points $p/q$. Investigating the intersection of this set with a suitable manifold comprises one of the most challenging problems in Diophantine approximation. It is known that the structure of this set, especially for large $\lambda$, depends on the manifold. For some manifolds it may be empty, while for others it may have relatively large Hausdorff dimension.

We will concentrate on the case of the Veronese curve $V_n$. We discuss, what is known about the Hausdorff dimension of the set $S_n(\lambda) \cap V_n$ and will talk about the recent results of the speaker and Bugeaud which impose new bounds on that dimension.

Minju Lee (Yale University)
2021-03-17 14:30 - 16:30 | https://stable.researchseminars.org/talk/DAHD-webinar/28/
Title: Orbit closures of unipotent flows for hyperbolic manifolds with Fuchsian ends. (https://stable.researchseminars.org/talk/DAHD-webinar/28/)
by Minju Lee (Yale University) as part of Webinar on Diophantine approximation and homogeneous dynamics

Abstract
This is joint work with Hee Oh. We establish an analogue of Ratner's orbit closure theorem for any connected closed subgroup generated by unipotent elements in $\mathrm{SO}(d,1)$ acting on the space $\Gamma\backslash\mathrm{SO}(d,1)$, assuming that the associated hyperbolic manifold $M=\Gamma\backslash\mathbb{H}^d$ is a convex cocompact manifold with
Fuchsian ends. For $d = 3$, this was proved earlier by McMullen, Mohammadi and Oh. In a higher dimensional case, the possibility of accumulation on closed orbits of intermediate subgroups causes serious issues, but in the end, all orbit closures of unipotent flows are relatively
homogeneous. Our results imply the following: for any $k\geq 1$,

(1) the closure of any $k$-horosphere in $M$ is a properly immersed submanifold;

(2) the closure of any geodesic $(k+1)$-plane in $M$ is a properly immersed submanifold;

(3) an infinite sequence of maximal properly immersed geodesic $(k+1)$-planes intersecting $\mathrm{core} M$ becomes dense in $M$.

Pablo Shmerkin (Torcuato Di Tella University)
2021-03-10 16:30 - 18:30 | https://stable.researchseminars.org/talk/DAHD-webinar/29/
Title: Beyond Furstenberg's intersection conjecture (https://stable.researchseminars.org/talk/DAHD-webinar/29/)
by Pablo Shmerkin (Torcuato Di Tella University) as part of Webinar on Diophantine approximation and homogeneous dynamics

Abstract
Hillel Furstenberg conjectured in the 1960s that the intersections of closed $\times 2$ and $\times 3$-invariant Cantor sets have "small" Hausdorff dimension. This conjecture was proved independently by Meng Wu and by myself; recently, Tim Austin found a simple proof. I will present some generalizations of the intersection conjecture and other related results.

Lifan Guan (Gottingen)
2021-04-14 13:30 - 15:30 | https://stable.researchseminars.org/talk/DAHD-webinar/30/
Title: Divergent trajectories on products of homogenous spaces (https://stable.researchseminars.org/talk/DAHD-webinar/30/)
by Lifan Guan (Gottingen) as part of Webinar on Diophantine approximation and homogeneous dynamics

Abstract
Thanks to Dani correspondence, it is now well-known that the set of singular vectors is closely related to the set of points with divergent trajectories in certain homogeneous dynamical systems. Since Yitwah Cheung's breakthrough work on the Hausdorff dimension of the set of 2-dim singular vectors, there have been lots of progress in singular vectors and divergent trajectories in the so-called "unweighted" cases. Otherwise, our understanding is quite limited. In this talk, I will mainly discuss the dimension formula for the set of divergent trajectories in products of "unweighted" homogeneous dynamical systems. This is a joint work with Jinpeng An, Antoine Marnat and Ronggang Shi.

Natalia Jurga (University of St Andrews)
2021-04-21 13:30 - 15:30 | https://stable.researchseminars.org/talk/DAHD-webinar/31/
Title: Random matrix products and self-projective sets (https://stable.researchseminars.org/talk/DAHD-webinar/31/)
by Natalia Jurga (University of St Andrews) as part of Webinar on Diophantine approximation and homogeneous dynamics

Abstract
A finite set of matrices $A \subset SL(2,R)$ acts on one-dimensional real projective space $RP^1$ through its linear action on $R^2$. In this talk we will be interested in the smallest closed subset of $RP^1$ which contains all attracting and neutral fixed points of matrices in $A$ and which is invariant under the projective action of $A$. Recently, Solomyak and Takahashi proved that if $A$ is uniformly hyperbolic and satisfies a Diophantine property, then the invariant set has Hausdorff dimension equal to the minimum of 1 and the critical exponent. In this talk we will discuss an extension of their result beyond the uniformly hyperbolic setting. This is based on joint work with Argyrios Christodoulou.

Pratyush Sarkar (Yale University)
2021-04-28 13:30 - 15:30 | https://stable.researchseminars.org/talk/DAHD-webinar/32/
Title: Generalization of Selberg's 3⁄16 theorem for convex cocompact thin subgroups of SO(n, 1) (https://stable.researchseminars.org/talk/DAHD-webinar/32/)
by Pratyush Sarkar (Yale University) as part of Webinar on Diophantine approximation and homogeneous dynamics

Abstract
Selberg’s 3/16 theorem for congruence covers of the modular surface is a beautiful theorem which has a natural dynamical interpretation as uniform exponential mixing. Bourgain-Gamburd-Sarnak's breakthrough works initiated many recent developments to generalize Selberg's theorem for infinite volume hyperbolic manifolds. One such result is by Oh-Winter establishing uniform exponential mixing for convex cocompact hyperbolic surfaces. These are not only interesting in and of itself but can also be used for a wide range of applications including uniform resonance free regions for the resolvent of the Laplacian, affine sieve, and prime geodesic theorems. I will present a further generalization to higher dimensions and some of these immediate consequences.

Henna Koivusalo (University of Bristol)
2021-05-05 13:30 - 15:30 | https://stable.researchseminars.org/talk/DAHD-webinar/33/
Title: Linear repetitivity in polytopal cut and project sets (https://stable.researchseminars.org/talk/DAHD-webinar/33/)
by Henna Koivusalo (University of Bristol) as part of Webinar on Diophantine approximation and homogeneous dynamics

Abstract
Cut and project sets are aperiodic point patterns obtained by projecting an irrational slice of the integer lattice to a subspace. One way of classifying aperiodic sets is to study the number and repetition of finite patterns. From this perspective, sets with patterns repeating linearly often, called linearly repetitive sets, can be viewed as the most ordered aperiodic sets. Repetitivity of a cut and project set depends on the slope and shape of the irrational slice. The cross-section of the slice is known as the window. In earlier works, joint with subsets of {Haynes, Julien, Sadun, Walton}, we showed that many properties of cut and project sets with a cube window can be studied in the language of Diophantine approximation. For example, linear repetitivity holds if and only if the following two conditions are satisfied: (i) the cut and project set has minimal number of different finite patterns (minimal complexity), and (ii) the irrational slope satisfies a badly approximable condition. In a new joint work with Jamie Walton, we give a generalisation of this result to all polytopal windows satisfying a mild geometric condition. A key step in the proof is a decomposition of the cut and project scheme, which allows us to make sense of condition (ii) for general polytopal windows.
The talk will cover motivation and history of studying cut and project sets, showcase a series of results on their repetitivity properties highlighting the number theory connections, and finish with the new results which move beyond Diophantine approximation.

Damaris Schindler (University of Göttingen)
2021-05-12 13:30 - 15:30 | https://stable.researchseminars.org/talk/DAHD-webinar/34/
Title: Density of rational points near/on compact manifolds with certain curvature conditions (https://stable.researchseminars.org/talk/DAHD-webinar/34/)
by Damaris Schindler (University of Göttingen) as part of Webinar on Diophantine approximation and homogeneous dynamics

Abstract
In this talk I will discuss joint work with Shuntaro Yamagishi where we establish an asymptotic formula for the number of rational points, with bounded denominators, within a given distance to a compact submanifold $M$ of $\mathbb{R}^n$ with a certain curvature condition. Technically we build on work of Huang on the density of rational points near hypersurfaces. One of our goals is to explore generalisations to higher codimension. In particular we show that assuming certain curvature conditions in codimension at least two, leads to upper bounds for the number of rational points on $M$ which are even stronger than what would be predicted by the analogue of Serre's dimension growth conjecture.

Nattalie Tamam (UC San Diego)
2021-05-19 15:30 - 17:30 | https://stable.researchseminars.org/talk/DAHD-webinar/35/
Title: When can we find a simple description of the divergence of trajectories? (https://stable.researchseminars.org/talk/DAHD-webinar/35/)
by Nattalie Tamam (UC San Diego) as part of Webinar on Diophantine approximation and homogeneous dynamics

Abstract
It is well known that the only singular numbers are rational numbers. Dani's correspondence ties this property to a simple algebraic description of divergent trajectories in $\mathrm{SL}_2(\mathbb{R})$ under the action of the diagonal group. Similar principles can be utilised to define obvious divergent trajectories in a more general setting. We will discuss the existence of non-obvious divergent trajectories under the action of different diagonal subgroups, and the diophantine meaning of their existence (or lack thereof). This is a joint work with Omri Nisan Solan.

Andrew Scoones (University of York)
2021-05-26 13:30 - 15:30 | https://stable.researchseminars.org/talk/DAHD-webinar/36/
Title: On the abc Conjecture in Algebraic Number Fields (https://stable.researchseminars.org/talk/DAHD-webinar/36/)
by Andrew Scoones (University of York) as part of Webinar on Diophantine approximation and homogeneous dynamics

Abstract
While the abc Conjecture remains open, much work has been done on weaker versions, and on generalising the conjecture to number fields. Stewart and Yu were able to give an exponential bound for $\max\{a, b, c\}$ in terms of the radical over the integers, while Györy was able to give an exponential bound for the projective height $H(a, b, c)$ in terms of the radical for algebraic integers. We generalise Stewart and Yu's method to give an improvement on Györy's bound for algebraic integers.

Anthony Poels and Damien Roy (University of Ottawa)
2021-10-08 14:00 - 16:00 | https://stable.researchseminars.org/talk/DAHD-webinar/37/
Title: Simultaneous rational approximation to successive powers of a real number (https://stable.researchseminars.org/talk/DAHD-webinar/37/)
by Anthony Poels and Damien Roy (University of Ottawa) as part of Webinar on Diophantine approximation and homogeneous dynamics

Abstract
Let $\xi\in\mathbb{R}\setminus\bar{\mathbb{Q}}$ be a real transcendental number and let $n$ be a positive integer. By pioneer work of Davenport
and Schmidt from 1969, we know that the exponent $\tau_{n+1}(\xi)$
of best approximation to $\xi$ by algebraic integers of degree
at most $n+1$ is at least equal to $1+1/\lambda_n(\xi)$, where
$\lambda_n(\xi)$ stands for the uniform exponent of rational
approximation to the successive powers $1,\xi,\dots,\xi^n$
of $\xi$. So any upper bound on $\lambda_n(\xi)$ which holds
for any $\xi\in\mathbb{R}\setminus\bar{\mathbb{Q}}$ provides a lower bound on
$\tau_{n+1}(\xi)$ which is also independent of $\xi$. In this talk,
we present new tools which yield, for each integer $n\ge 4$, a
significantly improved upper bound on $\lambda_n(\xi)$ and thus
a refined lower bound on $\tau_{n+1}(\xi)$. The new lower bound is
$n/2+a\sqrt{n}+4/3$ with $a=(1-\log(2))/2\simeq 0.153$, instead of the
current $n/2+\mathcal{O}(1)$.

As usual, the starting point is the sequence of so-called
minimal points $\mathbf{x}_1,\mathbf{x}_2,\mathbf{x}_3,\ldots$ in $\mathbb{Z}^{n+1}$ defined initially
by Davenport and Schmidt. Our strategy consists in estimating from above
the height of the subspaces of $\mathbb{R}^{n-\ell+1}$ generated by $n-\ell+1$
consecutive coordinates from each point among $\mathbf{x}_i,\mathbf{x}_{i+1},\dots,\mathbf{x}_q$
for given $i\le q$. To this end, we first need a lower bound for the
dimension of such spaces.

In the first part of the talk, we present the required background with some historical perspective and our key algebraic result concerning the dimension of the above mentioned spaces. In the second part, we look at their heights from different
perspectives and outline the general strategy of the proof.

Alex Gorodnik (University of Zurich)
2021-10-22 14:00 - 16:00 | https://stable.researchseminars.org/talk/DAHD-webinar/38/
Title: Products of linear forms and distribution of translated measures (https://stable.researchseminars.org/talk/DAHD-webinar/38/)
by Alex Gorodnik (University of Zurich) as part of Webinar on Diophantine approximation and homogeneous dynamics

Abstract
We explore the behavior of the counting function which represents the number of solutions
of a multiplicative Diophantine problem. The argument is based on analysis of measures translated under a group action on homogeneous spaces. Ultimately we explain how estimates on correlations of translated measures lead to a quantitative asymptotic formula for the counting function. This is a joint work with Björklund and Fregolli.

Jimmy Tseng (University of Exeter)
2021-11-05 14:30 - 16:30 | https://stable.researchseminars.org/talk/DAHD-webinar/39/
Title: Shrinking target and horocycle equidistribution (https://stable.researchseminars.org/talk/DAHD-webinar/39/)
by Jimmy Tseng (University of Exeter) as part of Webinar on Diophantine approximation and homogeneous dynamics

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.

Yitwah Cheung (Tsinghua University)
2021-11-19 14:30 - 16:30 | https://stable.researchseminars.org/talk/DAHD-webinar/40/
Title: Invariants of Diophantine Approximation (https://stable.researchseminars.org/talk/DAHD-webinar/40/)
by Yitwah Cheung (Tsinghua University) as part of Webinar on Diophantine approximation and homogeneous dynamics

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.

Jiyoung Han (Tata Institute)
2021-12-03 14:30 - 16:30 | https://stable.researchseminars.org/talk/DAHD-webinar/41/
Title: Higher moment formulas for Siegel transforms and applications to limit distributions of functions of counting lattice points (https://stable.researchseminars.org/talk/DAHD-webinar/41/)
by Jiyoung Han (Tata Institute) as part of Webinar on Diophantine approximation and homogeneous dynamics

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 and Södergren (2019).
This is joint work with Mahbub Alam and Anish Ghosh.

Demi Allen (University of Warwick)
2021-12-17 14:30 - 16:30 | https://stable.researchseminars.org/talk/DAHD-webinar/42/
Title: An inhomogeneous Khintchine-Groshev Theorem without monotonicity (https://stable.researchseminars.org/talk/DAHD-webinar/42/)
by Demi Allen (University of Warwick) as part of Webinar on Diophantine approximation and homogeneous dynamics

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 $\R^{nm}$ 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.
