Diophantine Approximation, Fractal Geometry and Related topics / Approximation diophantienne, géométrie fractale et sujets connexes

Liste des Contributions

1. Victor Beresnevich

03/06/2024 09:30

Khintchine’s Theorem - one hundred years on!

Khintchine’s Theorem (1924) on rational approximations to real numbers
is one of the most beautiful applications of the Borel-Cantelli lemma outside of probability theory. In this talk I will discuss problems and progress concerning inhomogeneous generalisations of this classical result, including the Duffin-Schaeffer Conjecture in the...

2. Sam Chow

03/06/2024 11:00

Multiplicative approximation on hypersurfaces

Diophantine approximation on manifolds has been a major theme for many decades and has seen remarkable recent progress. I discuss joint work in preparation
with Han Yu on the multiplicative analogue.

3. Evgeniy Zorin

03/06/2024 11:40

Exploring the Limits: Unbounded Diophantine Approx- imations and Matrix Transformations

In this talk, I will present our recent advancements on the shrinking target
problem of matrix transformation on tori and their subvarieties. For tori, we can provide sharp asymptotic results in a remarkably broad setting. This research has been naturally linked with expansion of the Mass Transference...

4. Nikolay Moshchevitin

03/06/2024 14:10

Inhomogeneous approximation revisited

We will discuss classical and modern results related to linear inhomogeneous
Diophantine approximation. We recall the history of the problem from Kronecker’s approximation theorem, transference theory developed by Khintchine, Jarník and Cassels to recent results concerning grids of lattices. In particular we explain a new theory of k-divergence...

5. Reynold Fregoli

03/06/2024 15:20

Improvements to Dirichlet’s Theorem in the multiplicative setup and equidistribution of averages along curves

In this talk, I will discuss uniform approximation by rationals vectors in
the multiplicative set-up. Curiously enough, in this context, Dirichlet’s Theorem is improvable, and, for m × n matrices the correct constant is bounded above by 2−m+1. One can also show that almost all...

6. Steven Robertson

03/06/2024 16:20

A Combinatorial Approach to the p(t)-adic Littlewood Conjecture

Let p be a prime and let p(t) be an irreducible polynomial with coefficients
in a field K. In 2004, de Mathan and Teuli ́e stated the p-adic Littlewood conjecture (p-LC) in analogy to the classical Littlewood conjecture. This talk focuses on the analogue of p-LC over the field of formal Laurent series with coefficients in K,...

7. Niclas Technau

03/06/2024 17:00

Counting Rational Points Near Manifolds

Choose your favourite, compact manifold M. How many rational points,
with denominator of bounded size, are near M? We report on joint work with Damaris Schindler and Rajula Srivastava addressing this question. Our new method reveals an intriguing interplay between number theory, harmonic analysis, and homogeneous dynamics.

9. Barak Weiss

04/06/2024 09:30

Singular vectors in manifolds, countable intersections, and Dirichlet spectrum

A vector x = (x1, ..., xd) in Rd is totally irrational if 1, x1, ..., xd are linearly
independent over rationals, and singular if for any ε > 0, for all large enough T, there are solutions p in Zd and q in {1, ..., T } to the inequality ∥qx − p∥ < εT −1/d. In previous work we showed that certain smooth manifolds...

10. Noy Soffer Aranov

04/06/2024 11:00

Hausdorff Dimension of the Set of Singular and Dirichlet Improvable Vectors
in Function Fields.

We compute the Hausdorff dimension of the set of singular vectors in
function fields and bound the Hausdorff dimension of the Dirichlet improvable vectors in this setting. Our results are a function field analog of the results of Cheung and Chevallier. This is part of joint work with Taehyeong Kim.

11. Erez Nesharim

04/06/2024 11:40

The Thue-Morse sequence has partial escape of mass over F2((1/t))

Every Laurent series in the field Fq ((1/t)) has a continued fraction expansion
whose digits are polynomials. De-Mathan and Teulie proved that the degrees of the partial quotients of the left shifts of every quadratic Laurent series are unbounded. Shapira and Paulin improved this by showing that, in fact, a positive...

12. Damien Roy

04/06/2024 14:10

Parametric geometry of numbers and simultaneous approximation to geometric
progressions

An important problem in Diophantine approximation is to determine, for
a given positive integer n, the supremum λ􏰖n of the exponents λ􏰖n(ξ) of uniform simultaneous rational approximation to geometric progressions (1, ξ, ξ2, . . . , ξn) whose ratio ξ is either a transcendental real number or an...

13. Stéphane Fischler

04/06/2024 15:20

Irrationality measures of values of E-functions

E-functions are a class of special functions introduced by Siegel in 1929;
they include the exponential and Bessel functions. Values of E-functions at algebraic numbers are never Liouville : they are never extremely well approximated by rationals. If an E-function with rational coefficients is evaluated at a rational number, a more precise...

14. Taehyeong Kim

04/06/2024 16:20

Infinitely badly approximable affine forms

In this talk, we will consider infinitely badly approximable affine forms in the
sense of inhomogeneous Diophantine approximation. We introduce a new concept of singularity for affine forms and characterize the infinitely badly approximable property by this singular property. We also discuss some applications of this characterization.

15. Agamemnon Zafeiropoulos

04/06/2024 17:00

A variant of Kaufman’s measures in two dimensions.

An old result of Kaufman showed that the set Bad of badly approximable
numbers supports a family of probability measures with polynomial decay rate on their Fourier transform. We show that the same phenomenon can be observed in a two-dimensional setup: we consider the set
B={(α,γ)∈[0,1]2 :inf∥qα−γ∥>0} and we prove that it supports certain...

16. Dmitry Kleinbock

05/06/2024 09:30

“Simultaneously dense and non-dense” orbits in homogeneous dynamics and
Diophantine approximation

Consider a non-compact homogeneous space X with the action of a diagonal
one-parameter subgroup. It is known that the set of points in X with bounded forward orbits has full Hausdorff dimension. Question: what about points with forward orbits both bounded and accumulating on a given z ∈ X? We...

17. Frédéric Paulin

05/06/2024 11:00

On Hausdorff dimension in inhomogeneous Diophantine approximation over
global function fields

We study inhomogeneous Diophantine approximation by elements of a
global function field (over a finite field) in its completion for a discrete valuation. Given an (m,n) matrix A with coefficients in this completion and a small r > 0, we obtain an effective upper bound for the Hausdorff dimension...

18. René Pfitscher

05/06/2024 12:10

Counting rational approximations in flag varieties

In the divergence case of Khintchine's theorem, Schmidt obtained an asymptotic formula for the number of rational approximations of bounded height to almost every real number. Using exponential mixing in the space of lattices, we prove versions of this theorem for intrinsic diophantine approximation on quadrics, grassmannians, and other...

19. Yann Bugeaud

06/06/2024 09:30

On the b-ary expansion of e

Let b ≥ 2 be an integer. The exponent vb (resp., vb′) and the uniform
′
rational numbers whose denominator is a power of b (resp., is of the form br(bs − 1)). Said differently and informally, we look at the lengths of the blocks of digit 0 (or of digit (b − 1)) and at the lengths of repeated blocks in the base-b expansion of a
exponent v􏰖b (resp., v􏰖 ) measure...

20. Stéphane Seuret

06/06/2024 11:00

An asymmetric version of the Littlewood conjecture

In this talk, we study an asymmetric version of the Littlewood
conjecture proposed by Y. Bugeaud. A parameter σ ∈ [0,1] being fixed, we study the set B(σ) of those pairs of real numbers (x,y) such thatinfq≥1(q · max(∥qx∥ ∥qy∥)1+σ min(∥qx∥ ∥qy∥)1−σ ) > 0. Counter-examples to the Littlewood conjecture would belong to B(0) and appear as an...

21. Volodymyr Pavlenkov

06/06/2024 11:40

Inhomogeneous Diophantine Approximation with restraint denominators on
M0-sets and some applications

This is a joint result with Evgeniy Zorin.
In this talk I will present a Schmidt-type theorem for Diophantine approximations with restraint denominators of sufficiently slow growth on M0-sets. Basically, the balance condition between the growth rate of denominators and the decay rate of...

22. Cagri Sert

06/06/2024 14:10

Projections of self-affine fractals

If a subset X of Rd is projected onto a linear subspace then the Hausdorff
dimension of its image is bounded above by the rank of the projection and by the dimension of the set X itself. When the Hausdorff dimension of the image is smaller than both of these values the projection is called an exceptional projection for the set X. By the classical theorem...

23. Gaurav Aggarwal

06/06/2024 15:20

Joint Equidistribution of Approximates

The distribution of integer points on varieties has occupied mathematicians
for centuries. In the 1950’s Linnik used an “‘ergodic method” to prove the equidistribution of integer points on large spheres under a congruence condition. As shown by Maaß, this problem is closely related to modular forms. Subsequently, there were spectacular developments...

24. Shreyasi Datta

06/06/2024 16:20

Rational points near manifolds and the Khintchine theorem

We discuss a problem in Diophantine approximation which is related to counting rational points near a manifold. The proof uses tools from homogeneous dynamics and geometry of numbers. This is a joint work with Victor Beresnevich.

25. Jiyoung Han

06/06/2024 17:00

Quantitative Khintchine--Groshev theorem on S-arithmetic numbers

In this talk, I would like to introduce two analogs of S-arithmetic generalization of Diophantine approximation problems. One way to obtain quantitative results for Diophantine approximation over the real field is by utilizing Schmidt's counting theorem on the family of expanding Borel sets. We will explore how this approach...

26. Anish Ghosh

07/06/2024 09:30

Dynamics on the space of affine lattices and inhomogeneous Diophantine
approximation.

We establish a new dynamical result on the space of affine lattices. Using
this, we uncover some new Diophantine properties of affine forms. Joint work with Gaurav Aggarwal.

27. Catalin Badea

07/06/2024 11:00

Times-2 and Times-3 Invariant Measures and Exceptional Sets of Uniform
Distribution

We explore Furstenberg’s times-2, times-3 conjecture, which poses the
question of whether the normalized Lebesgue measure is the sole atom-free probability measure invariant under both times-2 and times-3 maps. Additionally, we analyze the size of exceptional sets associated with (almost) uniform...

28. Prasuna Bandi

07/06/2024 11:40

Exact weighted approximation

I will discuss the results on the Exact ψ-approximable set in Diophantine
approximation. Further, in the weighted setting, we will show that its Hausdorff dimension is equal to that of the ψ-well approximable set under certain conditions on ψ. This is a joint work with Reynold Fregoli.

29. Stéphane Jaffard

07/06/2024 14:10

Some interplays between multifractal analysis and Diophantine approximation

Multifractal analysis deals with the determination of the pointwise
regularity of everywhere irregular functions. It is therefore not surprising that many functions which had been proposed as examples or counter-examples of “pathological” functions turned out to be multifractal. What is more remarkable is that they...

30. Arnaud Durand

07/06/2024 15:20

Capacities and (large) intersections for random sets in metric spaces, with
applications in dynamical Diophantine approximation

We consider, in general metric spaces, the classes of random sets that are
bound to intersect almost surely any deterministic set with positive capacity in a given gauge function. This property yields a lower bound on the size of the intersection of these random...

31. Edouard Daviaud

07/06/2024 16:20

A certain type of approximation by polynomials with algebraic coefficients

Let $N\in\mathbb{N}$ be an integer and $\mathcal{A}=\left\{q_1,....,q_N\right\}$ be a set of algebraic numbers. Given $k\in\mathbb{N}$ call $\mathcal{P}_{\mathcal{A},k}$ the set of polynomials of degree smaller than $k$ and coefficient in $\mathcal{A}$ by $\mathcal{P}_{\mathcal{A},k}$ and $\mathcal{P}_{\mathcal{A}}$...

32. Ioannis Tsokanos

07/06/2024 17:00

Stability and Shadowing of Non-invertible p-adic Dynamics

A continuous dynamical system is a couple (X,f) where (X,d) is a metric
space and f : X 􏰔→ X is a continuous map (called dynamic). The notions of stability and shadowing, introduced in the second third of the 20th century in the works of Andronov, Pontrjagin, Bowen, and Sinai, play a fundamental role in several branches of dynamical...

34. Diner de conférence / Conference diner

Place:

Les Noces de Jeannette
14, rue Favart - 75002 Paris

https://www.lesnocesdejeannette.com/

33. Dmitry Kleinbock -