Online Seminar in Diophantine Approximation and Related Topics
This seminar is purely online. Our talks are about Diophantine approximation and related topics, with speakers from all over the world. They are streamed live on Zoom.
Organizers :
- Antoine Marnat : antoine.marnat@u-pec.fr
- Nikolay Moshchevitin : nikolaus.moshchevitin@gmail.com
There are no fees, but registration is necessary. If you are interested in participating, please contact organizers per email. You can subscribe or unsubscribe to the mailing list. Registered users will receive an email before the talk with a link to the Zoom meeting. The seminar runs usually on Thursday at 12:00 GMT. Talks are 60 minutes and then time for questions.
Please make sure your audio is muted by default. You can ask questions in the chat or by unmuting the microphone and asking the speaker directly.
Next talks
Robert Frazer : Rajchman measures on generalizations of the Liouville numbers
Thursday, May 15th, 2025, 13:00 London / 14:00 Paris
Abstract: In 1980, Kaufman constructed a measure supported on the set of badly approximable numbers whose Fourier transform decays at a polynomial rate. In this talk, we will discuss a modification of Kaufman’s argument to construct Rajchman measures supported on sets of numbers having an infinite sequence of partial quotients satisfying a general type of condition. This generalizes a result of Bluhm, who constructed such measures on the set of Liouville numbers.
Dong Han Kim : Uniform Diophantine approximation on the Hecke group H_4
Thursday, May 22th, 2025, 13:00 London / 14:00 Paris
Abstract: Dirichlet's uniform approximation theorem is a fundamental result in Diophantine approximation that gives an optimal rate of approximation with a given bound.We study uniform Diophantine approximation properties on the Hecke group $\mathbf H_4$. For a given real number $\alpha$, we characterize the sequence of $\mathbf H_4$-bestapproximations of $\alpha$ and show that they are convergents of the Rosen continued fraction and the dual Rosen continued fraction of $\alpha$. We give analogous theorems of Dirichlet uniform approximation and the Legendre theorem with optimal constants.
This is joint work with Ayreena Bakhtawar and Seul Bee Lee.
Vasiliy Nekrasov : TBA
Thursday, June 5th, 2025, 13:00 London / 14:00 Paris
Based on https://arxiv.org/abs/2503.21180
Boaz Klartag: Lattice packing of spheres in high dimensions using a stochastically evolving ellipsoid
Thursday, June 19th, 2025, 13:00 London / 14:00 Paris
Abstract: We prove that in any dimension n there exists an origin-symmetric ellipsoid of volume c n^2 that contains no points of Z^n other than the origin. Here c > 0 is a universal constant. Equivalently, there exists a lattice sphere packing in R^n whose density is at least c n^2 / 2^n. Previously known constructions of sphere packings in R^n had densities of the order of magnitude of n / 2^n, up to logarithmic factors. Our proof utilizes a stochastically evolving ellipsoid that accumulates at least c n^2 lattice points on its boundary, while containing no lattice points in its interior except for the origin.
Previous talks
Timothée Bénard : Diophantine approximation and random walks on the modular surface
Thursday, April 3rd, 2025, 13:00 London / 14:00 Paris
Abstract : Khintchine's theorem is a key result in Diophantine approximation. Given a positive non-increasing function f defined over the integers, it states that the set of real numbers that are f-approximable has zero or full Lebesgue measure depending on whether the series of terms (f(n))_n converges or diverges. I will present a recent work in collaboration with Weikun He and Han Zhang in which we extend Khintchine's theorem to any self-similar probability measure on the real line. The argument involves the quantitative equidistribution of upper triangular random walks on SL_2(R)/SL_2(Z).
Matthias Gröbner : Equidistribution, covering radius, and Diophantine approximation
for rational points on the sphere
Thursday, March 27th, 2025, 13:00 London / 14:00 Paris
Abstract: This talk focuses on the counting and distribution of rational points on the sphere, with a particular emphasis on equidistribution in shrinking spherical caps. I will discuss connections to the covering radius problem and intrinsic Diophantine approximation. Based on joint work with Claire Burrin.
Manuel Hauke : Approximation by prime denominators: Twisted Diophantine approximation and approximation with chosen numerators
Thursday, March 13, 2025, 13:00 London / 14:00 Paris
Gaurav Aggarwal : Dimension bounds for singular affine forms
Thursday, March 6th 2025, 13:00 London / 14:00 Paris
Abstract: in this talk, I will establish upper bounds on the dimension of sets of singular affine forms in singly metric settings, where either the matrix or the shift is fixed. The results will be derived in a generalized weighted setup and for points sampled from fractals. This partially answers the questions posed by Das, Fishman, Simmons, Urbański, as well as by Kleinbock and Wadleigh.
This talk is based on https://arxiv.org/pdf/2501.01713
Video
Vanshika Jain : Relative Lonely Runner Spectra
Thursday, February 27th 2025, 13:00 London / 14:00 Paris
Abstract: The lonely runner conjecture states that for n runners on a unit-length track with constant, nonzero, integer speeds, all starting from the same position, there exists a time t when each runner is at least 1/(n+1) units away from the start line. This conjecture remains open for seven or more runners. For a given set of speeds, the maximum loneliness is defined to be the largest value L for which there is a time t at which every runner is at least L units away from the start line.
In this talk, I will introduce a related concept called the lonely runner spectra. Recent work by Noah Kravitz and Vikram Giri shows that these spectra possess a rich "hierarchical" structure. I will describe how these relative spectra exhibit rigid arithmetic properties and how each spectrum can be fully characterized by a finite computation. Finally, I will outline how such a computation can be used to completely characterize the maximum loneliness values for three runners up to any value strictly greater than zero. Based on joint work with Noah Kravitz.
Alon Agin : The Dirichlet spectrum
Thursday, February 13th 2025, 13:00 London / 14:00 Paris
Abstract: Akhunzhanov and Shatskov defined the Dirichlet spectrum, corresponding to mxn matrices and to norms on R^m and R^n. In case (m,n) = (2,1) and using the Euclidean norm on R^2, they showed that the spectrum is an interval. We generalize this result to arbitrary (m,n) with max(m,n)>1 and arbitrary norms, improving previous works from recent years. We also define some related spectra and show that they too are intervals. We also prove the existence of matrices exhibiting special properties with respect to their uniform exponent. Our argument is a modification of an argument of Khintchine from 1926.
Igor Pak : Counting trees and matching via continuous fractions
Thursday, January 30th, 2025, 17:00 London / 18:00 Paris / 9:00 Los Angeles
Abstract: In Combinatorics, a typical question asks to count the number of combinatorial objects of a certain kind, e.g. the number of spanning trees of perfect matchings in a given graph. In the past few years, the inverse question has also become popular, e.g. what is the smallest size graph which has a given number of spanning trees, or of a given number of perfect matchings? These questions turned out to be deeply related to classic problems and results in number theory.
In the first part of the talk I will give a brief overview of several combinatorial functions where this inverse problem has been resolved. I will then discuss a connection between two problems discussed above and Zaremba type questions and results on continued fractions. I will conclude with a discussion of our latest joint work with Chan and Kontorovich which gives best known bounds for spanning trees.
Nattalie Tamam and Shreyasi Datta : Weighted singular vectors for multiple weights
Thursday, January 23rd, 2025, 13:00 London / 14:00 Paris
Abstract : It follows from the Dirichlet theorem that every vector has `good' rational approximations. Singular vectors are the ones for which the Dirichlet theorem can be infinitely improved. An (obvious) example of singular vectors are the ones lying on rational hyperplanes. We will discuss the existence of totally irrational weighted singular vectors on manifolds, and also ones with high weighted uniform exponent. We will also mention some invariance of weighted uniform exponents in the case of manifolds. The talk is based on our joint work, see https://arxiv.org/abs/2409.17105.
Stéphane Fischler : Irrationality measures of values of E-functions
Thursday, 28 November 2024, 13:00 London / 14:00 Paris
Abstract: E-functions are a class of special functions introduced by Siegel in 1929; they include the exponential and Bessel functions. Very powerful qualitative results are known about their values, but their quantitative versions are more recent or still conjectural. In this lecture we shall focus on two recent results. First, values of E-functions at algebraic numbers are never Liouville : they are never extremely well approximated by rationals. Second, if an E-function with rational coefficients is evaluated at a rational number, a more precise result holds : if irrational, the value has exponent of irrationality 2, like a randomly chosen number. This is a joint work with Tanguy Rivoal, based mostly on results of Shidlovsky, Chudnovsky, André and Beukers.
Georgios Kotsovolis : BASS NOTE SPECTRA OF BINARY FORMS
Thursday, 21 November 2024, 13:00 London / 14:00 Paris
Abstract: pdf
Evgeniy Zorin : Rational points, manifolds, smoothness and Brownian Motion
Thursday, 7 November 2024, 13:00 London / 14:00 Paris
Abstract: pdf
Video
Nikita Shulga : Dirichlet improvability in L_p norms
Thursday, 24 October 2024, 13:00 London / 14:00 Paris
Abstract : For a norm $F$ on $R^2$, we consider the set of $F$-Dirichlet improvable numbers. In case of $F$ a supremum norm, it is well-known that the corresponding set of Drichlet improvable numbers is equal to the union of badly approximable numbers with rational numbers. It is also known that both badly approximable numbers and each $F$-Dirichlet improvable numbers are of measure zero and of full Hausdorff dimension.
Using the classification of critical lattices for unit balls in $L_p$, we provide a complete and effective characterization of Dirichlet improvable numbers with respect to $L_p$ norm in terms of the occurrence of patterns in regular continued fraction expansions.
As an application, we answer two open questions by Kleinbock and Rao by showing that the set of $p$-Dirichlet improvable numbers, which are not badly approximable, is of full Hausdorff dimension. Similarly, we show that the set difference of Dirichlet improvable numbers in Euclidean norm ($p=2$) minus Dirichlet improvable numbers in taxicab norm ($p=1$) and vice versa, are of full Hausdorff dimension.
This is a joint work with Nikolay Moshchevitin.
Video
Anthony Poëls : On approximation to a real number by algebraic numbers of bounded degree
Thursday, 10 October 2024, 13:00 London / 14:00 Paris
Abstract : In his seminal 1961 paper, Wirsing studied how well a given transcendental real number ξ can be approximated by algebraic numbers α of degree at most n for a given positive integer n, in terms of the so-called naive height H(α) of α. He showed that the exponent ω_n^*(ξ) which measures this quality of approximation is at least (n + 1)/2. He also asked if we could even have ω_n^*(ξ) ≥ n as it is generally expected. Since then, all improvements on Wirsing's lower bound were of the form n/2 + O(1) until Badziahin and Schleischitz showed in 2021 that ω_n^*(ξ) ≥ an for each n ≥ 4, with a = 1/√3 ≃ 0.577. In my talk, I will first present the background and ideas behind the proof of Wirsing's lower bound. Secondly, using a new approach that is partly inspired by parametric geometry of numbers, I will explain how we can obtain ω_n*(ξ) ≥ an for each n ≥ 2, with a = 1/(2 − log 2) ≃ 0.765.
Documents : Slides, Video
Gaétan Guillot : Approximation of linear subspaces by rational linear subspaces
Tuesday, 28 May 2024, 12:00 London / 13:00 Paris / 14:00 Israel
Abstract : We elaborate on a problem raised by Schmidt in 1967 : rational approximation of linear subspaces of $\mathbb{R}^n$ . In order to study the quality approximation of irrational numbers by rational ones, one can introduce the exponent of irrationality of a number. We can then generalize this notion in the framework of vector subspaces for the approximation of a subspace by so-called rational subspaces.
After briefly introducing the tools for constructing this generalization, I will present the different possible studies of this object. Finally I will explain how we can construct spaces with prescribed exponents.
Video
Ioannis Tsokanos: Randomization in the Josephus Problem
Tuesday, 21 May 2024, 8:00 Sao Paulo / 12:00 London / 13:00 Paris / 14:00 Israel
Abstract : see pdf,
Video
Dubi Kelmer: Counting and distribution of rational points on the sphere
Tuesday, 7 May 2024, 9:00 Boston / 14:00 in London / 15:00 in Paris / 16:00 in Israel
Abstract: In this talk I will describe some recent results regarding the distribution of rational points on the unit n dimensional sphere.
In particular I will show how one can give an optimal bound for the remainder when counting the number of rational points with a bounded denominator. I will also discuss related problems on counting rational points that fall in a fixed, or shrinking target set on the sphere.
Video
Mahbub Alam: Diophantine approximation by restricted rationals over number fields
Tuesday, 30 April 2024, 12:00 London / 13:00 Paris / 14:00 Israel
Abstract: A few generalizations of the classical diophantine approximation have seen considerable interest, such as approximation by restricted rationals and approximation in spaces other than Euclidean spaces. We consider a hybrid: diophantine approximation over a number field by rationals whose numerators and denominators are restricted to satisfy some congruence conditions. We will show that the adeles of a number field is a natural object in this context and the above problem can be realized as an adelic lattice point counting problem. Moment formulae for adelic lattice point counting functions, obtained recently by Kim, are then utilized to obtain a quantitative result in this context. This is analogous to a result due to Schmidt and extends a result by the author, Shucheng Yu and Anish Ghosh.
Video
Eric Gaudron : Diophantine approximation on spheres
Thursday, 25 April 2024, 12:00 in London / 13:00 in Paris / 14:00 in Israel
Abstract: Dirichlet’s approximation theorem on spheres consists in approaching a vector located on the unit n-sphere by another vector on this sphere having rational coordinates. We explain how to get such a result in general, even when the underlying quadratic form is no longer positive-definite. Our statements generalize and improve on earlier results by Kleinbock & Merrill (2015) and Moshchevitin (2017). The proofs rely on the quadratic Siegel’s lemma obtained by the author and Gaël Rémond (2017).
Video
Shigeki Akiyama : Discretized rotation problems and Salem numbers
Tuesday, 2d April 2024, 12:00 Paris / 20:00 Tokyo
Abstract: Described rotation serves a typical number theoretical problem: it seems easy and elementary but is notoriously difficult.
First, I will give an overview of this problem. Second, I will discuss a recent development on the beta expansion in Salem number base, which has an interesting connection to the problem of discretized rotation. This is a joint work with Hichri Hachem.
Video
Arijit Ganguly : Diophantine transference principle in positive characteristic
Tuesday, 26 March 2024, 08:00 in New York / 12:00 in London / 13:00 in Paris/ 14:00 in Israel /20:00 in Beijing
Abstract : pdf.
Video
Anurag Rao : Badly approximable grids and k-divergent lattices
Tuesday, 19 March 2024, 08:00 in New York / 12:00 in London / 13:00 in Paris/ 14:00 in Israel /20:00 in Beijing
Abstract: pdf.
Documents : Slides and Video
Tushar Das: Thermodynamic expansions for the Hausdorff dimension of continued fraction Cantor sets via transfer operator perturbation
Tuesday, 12 March 2024, 09:00 in Chicago / 10:00 in New York / 15:00 in London / 16:00 in Paris/ 17:00 in Israel /23:00 in Beijing
Abstract: Continued fractions have provided a natural playground for several advances in number theory, geometry, topology, dynamics, analysis, and probability theory. I will report on some dimension-theoretic research -- https://arxiv.org/a/das_t_4.html -- regarding the fascinating fractals that arise from studying continued fractions and point to vistas among their conformal cousins about which much less is known. The talk will be accessible to those whose interests intersect a convex combination of dynamics, fractal geometry and Diophantine approximation.
Documents : Slides and Video
Leonardo Colzani : Rate of convergence in an ergodic transformation
Tuesday, February 20, 2024, 08:00 in New York / 12:00 in London / 13:00 in Paris/ 14:00 in Israel /20:00 in Beijing
Abstract: We discuss the speed of convergence in the numerical integration with Weyl sums over Kronecker sequences in the torus.
Video
Dmitrii Gayfulin and Manuel Hauke : Hausdorff dimension estimates for Sudler products with positive lower bound
Thursday, February 15, 2024, 08:00 in New York / 12:00 in London / 13:00 in Paris/ 14:00 in Israel /20:00 in Beijing
Abstract: For $\alpha \in \mathbb{R}$ and $N \in \mathbb{N}$, the Sudler product at stage $N$ is defined as
$$P_N(\alpha) := \prod_{r=1}^{N} 2 \left\lvert \sin \pi r\alpha \right\rvert.$$
It is known that $\liminf_{N \to \infty}P_N(\alpha)=0$ whenever the sequence of partial quotients in the continued fraction expansion of $\alpha$ contains infinitely many digits greater than $6$. In fact, it was conjectured by Lubinsky that $\liminf$ equals zero for all real numbers. However, it was shown by Verschueren and independently by Grepstad, Kaltenböck and Neumüller that for $\alpha$ equal to the golden ratio $[0;1,1,1,\ldots]$ one has $\liminf_{N \to \infty}P_N(\alpha)>0$. Later, some other counterexamples were found, all of them also were quadratic irrationals. In a joint paper with Manuel Hauke, we show that $\liminf_{N \to \infty} P_N(\alpha) >0$ whenever the sequence of partial quotients in the continued fraction expansion of $\alpha$ exceeds $3$ only finitely often. Furthermore we deduce a non-trivial lower bound of the HD of the set of $\alpha$ satisfying $\liminf_{N \to \infty} P_N(\alpha) >0$.
Video
Prasuna Bandy : Hausdorff dimension of Exact-\psi approximable set
Thursday, December 14th 2023, 13:00 GMT = 08:00 New York/ 13:00 London / 14:00 Paris / 15:00 Israel / 21:00 Beijing
Abstract : pdf.
Video
Sam Chow : Dispersion and Littlewood’s conjecture
Thursday, December 7th 2023, 11:00 GMT = 06:00 in New York / 11:00 in London / 12:00 in Paris / 13:00 Israel / 19:00 in Beijing
Abstract: I’ll discuss some problems related to Littlewood’s conjecture in diophantine approximation, and the role hitherto played by discrepancy theory. I’ll explain why our new dispersion-theoretic approach should, and does, deliver stronger results. Our dispersion estimate is proved using Poisson summation and diophantine inequalities. Joint with Niclas Technau.
Baowei Wang : Mahler's question of intrinsic Diophantine approximation on triadic Cantor set: a divergence theory
Tuesday, November 28th 2023, 11:00 GMT = 11:00 in London / 12:00 in Paris / 13:00 Israel / 19:00 in Beijing / 06:00 in New York
Abstract: This talk is concerned about the metric theory of intrinsic Diophantine approximation on the triadic Cantor set, i.e. approximating the points in the triadic Cantor set by rationals inside the Cantor set. We give a full measure criterion, but the zero measure case is still unknown.
Video
Bryce Kerr : Metric theory of Weyl sums
Thursday, November 2nd 2023, 11:00 GMT = 11:00 in London / 12:00 in Paris / 13:00 Israel / 19:00 in Beijing / 07:00 in New York
Abstract: In this talk I'll describe joint work with Chen, Maynard and Shparlinski which obtains new results concerning the Hausdorff dimension of the set of large values of exponential sums. I'll describe two sorts of results:
1) Metric estimates for exponential sums over polynomials: This combines ideas of Cassels with estimates of Hooley for counting solutions to equations with polynomials and shows the set of large values of exponential sums has a Cantor-type structure.
2) Metric estimates for more general smooth functions: We make use of an iterated Van der Corput type oscillatory integral estimate to show a similar Cantor-type structure as in the polynomial case.
Video
Edouard Daviaud : Approximation by rectangles on (non necessary product) missing digit sets
Thursday, October 19th 2023, 12:00 GMT = 15:00 Israeli time / 13:00 in London / 14:00 in Paris / 08:00 in New York
Abstract : In this talk we will discuss the problem of weighted approximation on missing digit sets in R². When the missing digit set is a product, as explained in a note of Allen and Ward (see arXiv:2205.07570 ), such problems can be studied using the mass transference from rectangle to rectangle established by Wang and Wu (arXiv:1909.00924 ). We will explain how we deal with missing digit sets that are non products. In particular, given an ergodic measure m on the fractal, we will provide a formula for points approximable at different rate in x and y by the orbit under the underlaying IFS of a m-typical point.
Gerardo Gonzalez Robert : Measure and dimension of sets approximated by equidistributed sequences.
Tuesday, October 10th 2023, 14:00 Israel (+3GMT =14:00 Paris, =19:00 Beijing =07:00 New York)
Abstract: In this talk, we give conditions entailing full measure of sets of points in n dimensional Euclidean space that can be approximated (with respect to a given approximation function) by the terms of a uniformly distributed sequence. We also determined their Hausdorff dimension whenever such sets are null for certain approximation functions. Our work relies on previously known discrepancy estimates as well as recent papers by Wang and Wu (Mathematische Annalen (2021) 381:243–317) and Kleinbock and Wang (Advances in Mathematics 428 (2023) 109154) on ubiquitous systems defined on rectangles.
This is joint work with M. Hussain, N. Shulga, and B. Ward. See arXiv:2308.16603 [math.NT]
Video
Felipe Ramirez : Inhomogeneous approximation of systems of linear forms with primitivity constraints
Thursday, June 29th 2023, 13:00 GMT = 16:00 Israeli time / 14:00 in London / 15:00 in Paris / 09:00 in New York
Abstract : For an $n$-by-$m$ matrix $X$ and a point $y\in\mathbb{R}^m$, we seek solutions $(\mathbf{p},\mathbf{q})\in \mathbb{Z}^m\times\mathbb{Z}^n$ to the inequality $|\mathbf{q}X - \mathbf{p} - y| < \psi(|q|)$, where $\psi$ is some fixed function of the natural numbers. Following a framework introduced by Dani, Laurent, and Nogueira in 2015, we impose additional coprimality-type conditions on the entries in the $(n+m)$-vector $(\mathbf{p}, \mathbf{q})$. In this set up, Dani, Laurent, and Nogueira proved results analogous to a doubly-metric inhomogeneous Khintchine--Groshev theorem, and they asked: 1) whether the result could be made singly-metric; 2) whether the family of "coprimality conditions" they considered could be expanded; 3) whether a monotonicity assumption could be removed from their results. I will discuss this as well as recent work with Demi Allen where we address the three questions.
David Simmons : Exact dimensions of the prime continued fraction Cantor sets
Thursday, June 22nd 2023, 13:00 GMT = 16:00 Israeli time / 14:00 in London / 15:00 in Paris/ 09:00 in New York
Abstract : We study the exact Hausdorff and packing dimensions of the \emph{prime Cantor set}, $\Lambda_P$, which comprises the irrationals whose continued fraction entries are prime numbers. We prove that the Hausdorff measure of the prime Cantor set cannot be finite and positive with respect to any sufficiently regular dimension function, thus negatively answering a question of Mauldin (2013) for this class of dimension functions. By contrast, under a reasonable number-theoretic conjecture we prove that the packing measure of the conformal measure on the prime Cantor set is in fact positive and finite with respect to the dimension function $\psi(r) = r^\delta \log^{-2\delta}\log(1/r)$, where $\delta$ is the dimension (conformal, Hausdorff, and packing) of the prime Cantor set.
Video
Reynold Fregoli : Almost-sure estimates for sums of reciprocals of fractional parts
Tuesday, May 9th 2023, 12:00 GMT = 13:00 in London / 14:00 in Paris / 15:00 in Israel and Moscow
Abstract : pdf
Chris Smyth : Thue sets
Thursday, April 27th 2023, 12:00 GMT = 13:00 in London / 14:00 in Paris / 15:00 in Israel and Moscow
Abstract : Thue sets are countable closed subsets of the positive real line with specified extra properties. I discuss some naturally-occurring examples of these sets.
For the set L of Mahler measures of polynomials with integer coefficients, I discuss the (limited) evidence for the set L being a Thue set too.
Noah Kravitz : Lonely Runner spectra
Tuesday, April 25th 2023, 12:00 GMT = 13:00 in London / 14:00 in Paris / 15:00 in Israel and Moscow
Abstract: Dirichlet's Theorem says that for any real number t, there is some v in {1,2,...,n} such that tv is within 1/(n+1) of an integer. The Lonely Runner Conjecture of Wills and Cusick asserts that the constant1/(n+1)in this theorem cannot be improved by replacing {1,2,...,n} with a different set of n nonzero real numbers. The conjecture, although now more than 50 years old, remains wide open for n larger than 6. In this talk I will describe the "Lonely Runner spectra" that arise when one considers the "inverse problem" for the Lonely Runner Conjecture, and I will explain the (a priori surprising) "hierarchical" relations among these spectra. Based on joint work with Vikram Giri.
Files : slides, video
Oleg Karpenkov : On Hermite's problem, Jacobi-Perron type algorithms, and Dirichlet groups
Tuesday, March 28th 2023, 12:00 GMT = 13:00 in London / 14:00 in Paris / 15:00 in Israel and Moscow
Abstract : In this talk we introduce a new modification of the Jacobi-Perron algorithm in the three dimensional case.This algorithm is periodic for the case of totally-real conjugate cubic vectors. To the best of our knowledge this is the first Jacobi-Perron type algorithm for which the cubic periodicity is proven. This provides an answer in the totally-real case to the question of algebraic periodicity for cubic irrationalities posed in 1848 by Ch.Hermite.
We will briefly discuss a new approach which is based on geometry of numbers. In addition we point out one important application of Jacobi-Perron type algorithms to the computation of independent elements in the maximal groups of commuting matrices of algebraic irrationalities.
Simon Baker : Fractal measures and Number Theory
Thursday, March 23rd 2023, 14:00 CEST / 15:00 Israeli time
Abstract : Self-similar measures are among the most well studied examples of fractal measures. In this talk I will discuss their Diophantine properties, and the measure that they give to the set of normal numbers in a given base. This talk will be partly based upon a joint work with Amir Algom and Pablo Shmerkin, and partly based upon a joint work with Demi Allen, Sam Chow, and Han Yu.
Video
Roswitha Hofer : Exact order of discrepancy of normal numbers
Tuesday, February 28th 2023, 14:00 CEST / 15:00 Israel time
Abstract : pdf
Benjamin Ward : $\psi$-badly approximable points
Tuesday, January 17th 2023, 14:00 CEST / 15:00 Israel time
Abstract: In this talk I will discuss recent work with Henna Koivusalo, Jason Levesley, and Xintian Zhang on the set of $\psi$-badly approximable points. $\psi$-badly approximable points are those which are $\psi$-well approximable, but at the same time not $c\psi$-well approximable for arbitrary small constant $c>0$. In 2003 Bugeaud proved in the one dimensional setting that the Hausdorff dimension of $\psi$-badly approximable points is the same as the Hausdorff dimension of $\psi$-well approximable points. Our main result provides a partial $d$-dimensional analogue of Bugeaud'sresult. In order to do this we construct a Cantorset that simultaneously
captures the well approximable and badly approximable nature of $\psi$-badly approximable points.
Taehyeong Kim : A lower bound on the Hausdorff dimension of weighted singular vectors
Tuesday, January 10th 2023, 14:00 CEST / 15:00 Israel time
Abstract: Let w=(w_1, . . . , w_d) be an ordered d-tuple of positive real numbers such that w_1+...+w_d=1 and w_1 \geq ... \geq w_d. A
d-dimensional vector (x_1, . . . , x_q) in R^d is said to be w-singular if for every epsilon for all large enough T there are solutions p in Z^d and q in {1,...,T} such that |qx_i - p_i| < epsilon T^{-w_i} for all i. It was shown by Liao, Shi, Solan, and Tamam that the Hausdorff dimension of 2-dimensional weighted singular vectors is 2-1/(1+w_1). In this talk, we discuss a lower bound of the Hausdorff dimension of d-dimensional weighted singular vectors. This is a joint work with Jaemin Park.
Shreyasi Datta : Singular vectors in affine subspaces
Tuesday, November 22th 2022, 14:00 CEST / 15:00 Israel time
On a slightly different direction I want to briefly talk about about some simple yet surprising results on ψ-Dirichlet numbers and singular vectors in function field. The talk will be based on joint works with Yewei Xu.
Barak Weiss : Singular vectors in manifolds and countable intersections
Tuesday, November 15th 2022, 14:00 CEST / 15:00 Israel time
Abstract: A vector x = (x_1, ..., x_d) in R^d is totally irrational if 1, x_1, ..., x_d are linearly independent over rationals, and singular if for any epsilon, for all large enough T, there are solutions p in Z^d and q in {1, ..., T} to the inequality
||qx - p || < epsilon T^{-1/d}
In previous work we showed that certain smooth manifolds of dimension at least two, and certain fractals, contain totally irrational singular vectors. The argument for proving this is a variation on an old argument employed by Khintchine and Jarník. We now adapt this argument to show that for certain families of maps f_i: R^d -> R^{n_i}, certain manifolds contain points x such that f_i(x) is a singular vector for all i. This countable intersection property is motivated by some problems in approximation of vectors by vectors with coefficient in a number field. Joint work with Dmitry Kleinbock, Nikolay Moshchevitin and Jacqueline Warren.
Dzmitry Badziahin : Continued fractions of cubic irrationals
Tuesday, October 25th 2022, 14:00 CEST / 15:00 Israelian time
Abstract: It was discovered by Gauss that for any $r\in\mathbb{Q}$ the Laurent series of the function $(1+t)^r$ has an easy-to-describe continued fraction expansion. Later, A. Baker used the convergents of that fraction to produce the first effective upper bounds of the irrationality exponent of some algebraic numbers, including $\sqrt[3]{2}$, and thereby improved the classical result of Liouville for them. Later, his method was refined by many mathematicians, including Chudnovsky brothers, Rickert and Bennet. However, it only works for algebraic numbers of the form $\big(1+\frac{a}{N}\big)^r$. In this talk, I will show that there are many other cubic irrational Laurent series, apart from $(1+t)^r$, that enjoy a nice continued fraction expansion. We will see that there are many enough of them, so that their specializations cover all cubic irrational numbers and for at least some of them we can provide non-trivial upper bounds of their irrationality exponents.
Damien Roy : Diophantine approximation with constraints
Thursday, October 20th 2022, 15:00 CEST / 16:00 Israel time
Abstract: Following Schmidt, Thurnheer and Bugeaud-Kristensen, we study how Dirichlet’s theorem on linear forms needs to be modified when one requires that the vectors of coefficients of the linear forms make an acute angle at most \theta_0 with a fixed proper non-zero subspace V of R^n for a fixed \theta_0 \in (0,\pi/2). Assuming that the point of R^n that we approximating has linearly independent coordinates over Q, we obtain best possible exponents of approximation which surprisingly depend only on the dimension of V. Our estimates are derived by reduction to a result of Thurnheer while their optimality follows from a new general construction in parametric geometry of numbers involving angular constraints. (Joint work with Jeremy Champagne).
Reynold Fregoli : Multiplicatively badly approximable vectors
Tuesday, October 4th 2022, 14:00 CEST / 15:00 Israelian time
Abstract: The Littlewood Conjecture states that for all pairs of real numbers (α, β) the product |q||qα + p1||qβ + p2|
becomes arbitrarily close to 0 when the vector (q,p1,p2) ranges in Z^3 and q ≠ 0. To date, despite much progress, it is not known whether this statement is true. In this talk, I will discuss a partial converse of the Littlewood Conjecture, where the factor |q| is replaced by an increasing function f(|q|). More specifically, following up on the work of Badziahin and Velani, I will be interested in determining functions f for which the above product and its higher dimensional generalizations stay bounded away from 0 for at least one pair (α, β) ∈ R^2. I will show how this problem can be reduced to counting lattice points in certain distorted boxes, which will, in turn, require careful estimates of the minimum of the lattice and of its dual.
Johannes Schleischitz : Dirichlet spectrum for simultaneous approximation and a linear form
Tuesday, September 27th 2022, 14:00 CEST / 15:00 Israel time
Abstract: The Dirichlet spectrum is analogously defined to the famous Lagrange spectrum, but with respect to uniform approximation. We determine the Dirichlet spectrum, for simultaneous approximation as well as for the dual problem of approximation with a linear form, in Euclidean space of any dimension at least 2. It turns out it is as large as it can possibly be, that is it equals the entire interval [0,1]. We also present several refined claims, including on metrical theory and fractal settings. Proof ideas are sketched.
Nicolas de Saxce : Rational approximations to linear subspaces
Tuesday, September 13th 2022, 14:00 CEST / 15:00 Israel time
Abstract: Using diagonal orbits on the space of lattices, we revisit some old questions of Schmidt concerning Diophantine approximation on Grassmanian varieties, and in particular we prove a version of Dirichlet's principle in that setting.