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Documents : Virtualconference  | enregistrements trouvés : 141

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Odd symplectic Grassmannians are a family of quasi-homogeneous varieties with properties nevertheless similar to those of homogeneous spaces, such as the existence of a Schubert-type cohomology basis. In this talk based on joint work with Ryan Shifler, I will explain how to construct their curve neighbourhoods. Curve neighbourhoods were first introduced by Buch, Chaput, Mihalcea and Perrin in the homogeneous setting: it is the union of all rational curves of fixed degree passing through a given Schubert variety. Potential applications include the computation of minimal degrees in quantum cohomology.
Odd symplectic Grassmannians are a family of quasi-homogeneous varieties with properties nevertheless similar to those of homogeneous spaces, such as the existence of a Schubert-type cohomology basis. In this talk based on joint work with Ryan Shifler, I will explain how to construct their curve neighbourhoods. Curve neighbourhoods were first introduced by Buch, Chaput, Mihalcea and Perrin in the homogeneous setting: it is the union of all ...

14N35 ; 14N15 ; 14M15

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Given a representation V of a reductive group G, Braverman-Finkelberg-Nakajima defined a Poisson variety called the Coulomb branch, using a convolution algebra construction. This variety comes with a natural deformation quantization, called a Coulomb branch algebra. Important cases of these Coulomb branches are (generalized) affine Grassmannian slices, and their quantizations are truncated shifted Yangians.
Motivated by the geometric Satake correspondence and the theory of symplectic duality/3d mirror symmetry, we expect a categorical g-action on modules for these truncated shifted Yangians. I will explain three results in this direction. First, we have an indirect realization of this action, using equivalences with KLRW-modules. Second, we have a geometric relation between these generalized slices by Hamiltonian reduction. Finally, we have an algebraic version of this Hamiltonian reduction which we are able to relate to the first realization.
Given a representation V of a reductive group G, Braverman-Finkelberg-Nakajima defined a Poisson variety called the Coulomb branch, using a convolution algebra construction. This variety comes with a natural deformation quantization, called a Coulomb branch algebra. Important cases of these Coulomb branches are (generalized) affine Grassmannian slices, and their quantizations are truncated shifted Yangians.
Motivated by the geometric Satake ...

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In the framework of the study of K-theoretical Coulomb branches, Finkelberg-Tsymbaliuk introduced remarkable new algebras, the shifted quantum affine algebras an their truncations, in the spirit of the shifted Yangians of Brundan-Kleshev, Braverman-Finkelberg Nakajima, Kamnitzer-Webster-Weekes-Yacobi... We discuss the following points in representation theory of (truncated) shifted quantum affine algebras that we relate to representations of Borel quantum affine algebras by induction and restriction functors. We establish that the Grothendieck ring of the category of finite-dimensional representations has a natural cluster algebra structure. We propose a conjectural parametrization of simple modules of a non simply-laced truncation in terms of the Langlands dual quantum affine Lie algebra. We have several evidences, including a general result for simple finite-dimensional representations proved by using the Baxter polynomiality of quantum integrable models.
In the framework of the study of K-theoretical Coulomb branches, Finkelberg-Tsymbaliuk introduced remarkable new algebras, the shifted quantum affine algebras an their truncations, in the spirit of the shifted Yangians of Brundan-Kleshev, Braverman-Finkelberg Nakajima, Kamnitzer-Webster-Weekes-Yacobi... We discuss the following points in representation theory of (truncated) shifted quantum affine algebras that we relate to representations of ...

17B37 ; 17B10 ; 82B23 ; 13F60

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In this talk I will report on recent progress on two different problems in discrepancy theory.In the first part I will present a recent extension of the notion of jittered sampling to arbitrary partitions of the unit cube. In this joint work with Markus Kiderlen from Aarhus, we introduce the notion of a uniformly distributed triangular array. Moreover, we show that the expected Lp-discrepancy of a point sample generated from an arbitrary equi volume partition of the unit cube is always strictly smaller than the expected Lp-discrepancy of a set of N uniform random samples for p > 1.
The second part of the talk is dedicated to greedy energy minimization. I will give a new characterisation of the classical van der Corput sequence in terms of a minimization problem and will discuss various related open questions.
In this talk I will report on recent progress on two different problems in discrepancy theory.In the first part I will present a recent extension of the notion of jittered sampling to arbitrary partitions of the unit cube. In this joint work with Markus Kiderlen from Aarhus, we introduce the notion of a uniformly distributed triangular array. Moreover, we show that the expected Lp-discrepancy of a point sample generated from an arbitrary equi ...

11K38 ; 11K31 ; 52C25

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In this talk I will report on recent progress on two different problems in discrepancy theory. In the first part I will present a recent extension of the notion of jittered sampling to arbitrary partitions of the unit cube. In this joint work with Markus Kiderlen from Aarhus, we introduce the notion of a uniformly distributed triangular array. Moreover, we show that the expected Lp-discrepancy of a point sample generated from an arbitrary equi volume partition of the unit cube is always strictly smaller than the expected Lp-discrepancy of a set of N uniform random samples for p > 1.
The second part of the talk is dedicated to greedy energy minimization. I will give a new characterisation of the classical van der Corput sequence in terms of a minimization problem and will discuss various related open questions.
In this talk I will report on recent progress on two different problems in discrepancy theory. In the first part I will present a recent extension of the notion of jittered sampling to arbitrary partitions of the unit cube. In this joint work with Markus Kiderlen from Aarhus, we introduce the notion of a uniformly distributed triangular array. Moreover, we show that the expected Lp-discrepancy of a point sample generated from an arbitrary equi ...

11K38 ; 11K31 ; 52C25

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Virtualconference  On S-Diophantine Tuples
Ziegler, Volker (Auteur de la Conférence) | CIRM (Editeur )

Given a finite set of primes $S$ and a m-tuple $(a_{1},...,a_{m})$ of positive, distinct integers we call the m-tuple $S$-Diophantine, if for each 1 ≤ i < j ≤ m the quantity $a_{i}a_{j}+1$ has prime divisors coming only from the set $S$. In this talk we discuss the existence of m-tuples if the set of primes $S$ is small. We will discuss recent results concerning the case that $|S| = 2$ and $|S| = 3$.

11D61 ; 11Y50 ; 11A51

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Virtualconference  Poisson-generic points
Weiss, Benjamin (Auteur de la Conférence) | CIRM (Editeur )

I will discuss a criterion for randomness of sequences of zeros and ones which is strictly stronger than normality, butholds for almost every sequence generated by i.i.d. random variables with distribution {1/2, 1/2}. Briefly put, the idea is count the number of times blocks of length n appear in the initial block of length $2^n$. I will also discuss an extension of this idea to toral automorphisms. (joint work with Yuval Peres)

11K16 ; 37D99 ; 60F99

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We introduce a class of so-called "Ratner-Marklof-Strombergsson measures". These are probability measures supported on cut-and-project sets in Euclidean space of dimension d>1 which are invariant and ergodic for the action of the groups ASL_d(R) or SL_d(R) (affine or linear maps preserving orientation and volume). We classify the measures that can arise in terms of algebraic groups and homogeneous dynamics. Using the classification, we prove analogues of results of Siegel, Weil and Rogers about a Siegel summation formula and identities and bounds involving higher moments. We deduce results about asymptotics, with error estimates, of point-counting and patchcounting for typical cut-and-project sets. Joint work with Rene Ruehr and Yotam Smilansky.
We introduce a class of so-called "Ratner-Marklof-Strombergsson measures". These are probability measures supported on cut-and-project sets in Euclidean space of dimension d>1 which are invariant and ergodic for the action of the groups ASL_d(R) or SL_d(R) (affine or linear maps preserving orientation and volume). We classify the measures that can arise in terms of algebraic groups and homogeneous dynamics. Using the classification, we prove ...

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In a recent paper, Istvan Gaal and Laszlo Remete studied the integer solutions to binary quartic Thue equations of the form $x^4-dy^4 = \pm 1$, and used their results to determine pure quartic number fields which contain a power integral basis. In our talk, we propose a new way to approach this diophantine problem, and we also show how an effective version of the abc conjecture would allow for even further improvements. This is joint work with M.A. Bennett. We also discuss a relation between this quartic diophantine equation to recent joint work with P.-Z. Yuan.
In a recent paper, Istvan Gaal and Laszlo Remete studied the integer solutions to binary quartic Thue equations of the form $x^4-dy^4 = \pm 1$, and used their results to determine pure quartic number fields which contain a power integral basis. In our talk, we propose a new way to approach this diophantine problem, and we also show how an effective version of the abc conjecture would allow for even further improvements. This is joint work with ...

11D25 ; 11D57 ; 11R16

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

37B10 ; 37A30 ; 11K50 ; 28A80

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In this short course, we will discuss the problem of ranking with partially observed pairwise comparisons in the setting of Bradley-Terry-Luce (BTL) model. There are two fundamental problems: 1) top-K ranking, which is to select the set of K players with top performances; 2) total ranking, which is to rank the entire set of players. Both ranking problems find important applications in web search, recommender systems and sports competition.
In the first presentation, we will consider the top-K ranking problem. The statistical properties of two popular algorithms, MLE and rank centrality (spectral ranking) will be precisely characterized. In terms of both partial and exact recovery, the MLE achieves optimality with matching lower bounds. The spectral method is shown to be generally sub-optimal, though it has the same order of sample complexity as the MLE. Our theory also reveals the essentially only situation when the spectral method is optimal. This turns out to be the most favorable choice of skill parameter given the separation of the two groups.
The second presentation will be focused on total ranking. The problem is to find a permutation vector to rank the entire set of players. We will show that the minimax rate of the problem with respect to the Kendall’s tau loss exhibits a transition between an exponential rate and a polynomial rate depending on the signal to noise ratio of the problem. The optimal algorithm consists of two stages. In the first stage, games with very high or low scores are used to partition the entire set of players into different leagues. In the second stage, games that are very close are used to rank the players within each league. We will give intuition and some analysis to show why the algorithm works optimally.
In this short course, we will discuss the problem of ranking with partially observed pairwise comparisons in the setting of Bradley-Terry-Luce (BTL) model. There are two fundamental problems: 1) top-K ranking, which is to select the set of K players with top performances; 2) total ranking, which is to rank the entire set of players. Both ranking problems find important applications in web search, recommender systems and sports competition.
In ...

62C20 ; 62F07 ; 62J12

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We introduce a family of block-additive automatic sequences, that are obtained by allocating a weight to each couple of digits, and defining the nth term of the sequence as being the total weight of the integer n written in base k. Under an additional combinatorial difference condition on the weight function, these sequences can be interpreted as generalised Rudin-Shapiro sequences. We prove that these sequences have the same two-term correlations as sequences of symbols chosen uniformly and independently at random. The speed of convergence is independent of the prime factor decomposition of k. This extends work by E. Grant, J. Shallit, T. Stoll, and by P.-A. Tahay.
We introduce a family of block-additive automatic sequences, that are obtained by allocating a weight to each couple of digits, and defining the nth term of the sequence as being the total weight of the integer n written in base k. Under an additional combinatorial difference condition on the weight function, these sequences can be interpreted as generalised Rudin-Shapiro sequences. We prove that these sequences have the same two-term c...

11A63 ; 11K31 ; 68R15

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We consider two common pseudorandom number generators constructed from iterations of linear and Möbius maps
$x \mapsto gx$ and $ x \mapsto (ax+b)/(cx+d)$
over a residue ring modulo an integer q ≥ 2, which are known as congruential and inversive generators, respectively. There is an extensive literature on the pseudorandomness of elements $u_{n}, n=1,2,...$, of the corresponding orbits. In this talk we are interested in what happens in these orbits at prime times, that is, we study elements $u_{p}$, $p = 2, 3, . . .$, where $p$ runs over primes.
We give a short survey of previous results on the distribution of $u_{p}$ for the above maps and then:
- Explain how B. Kerr, L. Mérai and I. E. Shparlinski (2019) have used a method of N. M. Korobov (1972) to study the congruential generator on primes modulo a large power of a fixed prime, e.g. $q=3^{\gamma }$ with a large $\gamma$. We also give applications of this result to digits of Mersenne numbers $2^{p}-1$.
- Present a result of L. Mérai and I. E. Shparlinski (2020) on the distribution of the inversive generator on primes modulo a large prime, q. The proof takes advantage of the flexibility of Heath-Brown’s identity, while Vaughan’s identity does not seem to be enough for our purpose. We also pose several open questions and discuss links to Sarnak’s conjecture on pseudorandomness of the Möbius function.
We consider two common pseudorandom number generators constructed from iterations of linear and Möbius maps
$x \mapsto gx$ and $ x \mapsto (ax+b)/(cx+d)$
over a residue ring modulo an integer q ≥ 2, which are known as congruential and inversive generators, respectively. There is an extensive literature on the pseudorandomness of elements $u_{n}, n=1,2,...$, of the corresponding orbits. In this talk we are interested in what happens in these ...

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Zaremba's conjecture belongs to the area of continued fractions. It predicts that for any given positive integer q there is a positive a, a < q, (a,q)=1 such that all partial quotients b_j in its continued fractions expansion a/q = 1/b_1+1/b_2 +... + 1/b_s are bounded by five. At the moment the question is widely open although the area has a rich history of works by Korobov, Hensley, Niederreiter, Bourgain and many others. We survey certain results concerning this hypothesis and show how growth in groups helps to solve different relaxations of Zaremba's conjecture. In particular, we show that a deeper hypothesis of Hensley concerning some Cantor-type set with the Hausdorff dimension >1/2 takes place for the so-called modular form of Zaremba's conjecture.
Zaremba's conjecture belongs to the area of continued fractions. It predicts that for any given positive integer q there is a positive a, a 1/2 takes place for the so-called modular form of Zaremba's conjecture....

11A55 ; 11J70 ; 11B30 ; 20G05 ; 20G40

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In the lecture we prove a lower estimate for the average of the absolute value of the remainder term of the prime number theorem which depends in an explicit way on a given zero of the Riemann Zeta Function. The estimate is only interesting if this hypothetical zero lies off the critical line which naturally implies the falsity of the Riemann Hypothesis. (If the Riemann Hypothesis is true, stronger results areobtainable by other metods.) The first explicit results in this direction were proved by Turán and Knapowski in the 1950s, answering a problem of Littlewood from the year 1937. They used the power sum method of Turán. Our present approach does not use Turán’s method and gives sharper results.
In the lecture we prove a lower estimate for the average of the absolute value of the remainder term of the prime number theorem which depends in an explicit way on a given zero of the Riemann Zeta Function. The estimate is only interesting if this hypothetical zero lies off the critical line which naturally implies the falsity of the Riemann Hypothesis. (If the Riemann Hypothesis is true, stronger results areobtainable by other metods.) The ...

11M26 ; 11N05 ; 11N30

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For any fixed coprime positive integers a, b and c with min{a, b, c} > 1, we prove that the equation $a^{x}+b^{y}=c^{z}$ has at most two solutions in positive integers x, y and z, except for one specific case which exactly gives three solutions. Our result is essentially sharp in the sense that there are infinitely many examples allowing the equation to have two solutions in positive integers. From the viewpoint of a well-known generalization of Fermat’s equation, it is also regarded as a 3-variable generalization of the celebrated theorem of Bennett [M.A.Bennett, On some exponential equations of S.S.Pillai, Canad. J. Math. 53(2001), no.2, 897-922] which asserts that Pillai’s type equation $a^{x}-b^{y}=c$ has at most two solutions in positive integers x and y for any fixed positive integers a, b and c with min {a, b} > 1. In this talk we give a brief summary of corresponding earlier results and present the main improvements leading to this definitive result. This is a joint work with T. Miyazaki.
For any fixed coprime positive integers a, b and c with min{a, b, c} > 1, we prove that the equation $a^{x}+b^{y}=c^{z}$ has at most two solutions in positive integers x, y and z, except for one specific case which exactly gives three solutions. Our result is essentially sharp in the sense that there are infinitely many examples allowing the equation to have two solutions in positive integers. From the viewpoint of a well-known generalization of ...

11D61 ; 11D41 ; 11D45

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Virtualconference  Bertini and Northcott
Pazuki, Fabien (Auteur de la Conférence) | CIRM (Editeur )

I will report on joint work with Martin Widmer. Let X be a smooth projective variety over a number field K. We prove a Bertini-type theorem with explicit control of the genus, degree, height, and field of definition of the constructed curve on X. As a consequence we provide a general strategy to reduce certain height and rank estimates on abelian varieties over a number field K to the case of Jacobian varieties defined over a suitable extension of K. We will give examples where the strategy works well!
I will report on joint work with Martin Widmer. Let X be a smooth projective variety over a number field K. We prove a Bertini-type theorem with explicit control of the genus, degree, height, and field of definition of the constructed curve on X. As a consequence we provide a general strategy to reduce certain height and rank estimates on abelian varieties over a number field K to the case of Jacobian varieties defined over a suitable extension ...

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The theory for trend filtering has been developed in a series of papers: Tibshirani [2014], Wang et al. [2016], Sadhanala and Tibshirani [2019], Sadhanala et al. [2017], Guntuboyina et al. [2020], and see Tibshirani [2020] for a very good overview and further references. In this course we will combine the approach of Dalalyan et al. [2017] with dual certificates as given in Candes and Plan [2011]. This allows us to obtain new results and extensions.

The trend filtering problem in one dimension is as follows. Let $Y\sim \mathcal{N}(f^{0},\ I)$ be a vector of observations with unknown mean $f^{0} :=\mathrm{E}Y$. Let for $f\in \mathbb{R}^{n},$
$$
(\triangle f)_{i}\ :=\ f_{i}-f_{i-1},\ i\geq 2,
$$
$$
(\triangle^{2}f)_{i}\ :=\ f_{i}-2f_{i-1}+f_{i-2},\ i\geq 3,
$$
$$
(\triangle^{k}f)_{i}\ :=\ (\triangle(\triangle^{k-1}f))_{i},\ i\geq k+1.
$$
Then we consider the estimator
$$
\hat{f}\ :=f\min_{\in \mathbb{R}^{n}}\{\Vert Y-f\Vert_{2}^{2}/n+2\lambda\Vert\triangle^{k}f\Vert_{1}\}.
$$
Let $S_{0} :=\{j\ :\ (\triangle^{k}f^{0})_{j}\neq 0\}$ and $s_{0} :=|S_{0}|$ its size. We want to prove a "oracle'' type of result: for an appropriate choice of the tuning parameter, and modulo $\log$-factors, with high probability $\Vert\hat{f}-f^{0}\Vert_{2}^{2}/n\lesssim(s_{0}+1)/n$. For this one may apply the general theory for the Lasso. Indeed, the above is a Lasso estimator if we write $f=\Psi b$ where $\Psi\in \mathbb{R}^{n\times n}$ is the "design matrix'' or "dictionary'' and where $b_{j}=(\triangle^{k}f)_{j}, j\geq k+1$. We present the explicit expression for this dictionary and then will notice that the restricted eigenvalue conditions that are typically imposed for Lasso problems do not hold. What we will do instead is use a "dual certificate'' $q$ with index set $\mathcal{D} :=\{k+1,\ .\ .\ .\ ,\ n\}$. We require that $q_{j}=\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(\triangle^{k}f^{0})_{j}$ if $j\in S_{0}$ and such that $|q_{j}|\leq 1-w_{j}$ if $j\in \mathcal{D}\backslash S_{0}$, where $\{w\}_{j\in D\backslash S_{0}}$ is a given set of noise weights. Moreover, we require $q_{k+1}=q_{n}=0.$ We call such $q$ an interpolating vector. We show for an appropriate choice of $\lambda$
$$
\Vert\hat{f}-f^{0}\Vert_{2}^{2}/n<\sim(s_{0}+1)/n+n\lambda^{2}\Vert\triangle^{k}q\Vert_{2}^{2}
$$
and that the two terms on the right hand side can be up to $\log$ terms of the same order if the elements in $S_{0}$ corresponding to different signs satisfy a minimal distance condition.

We develop a non-asymptotic theory for the problem and refine the above to sharp oracle results. Moreover, the approach we use allows extensions to higher dimensions and to total variation on graphs. For the case of graphs with cycles the main issue is to determine the noise weights, which can be done by counting the number of times an edge is used when traveling from one node to another. Extensions to other loss functions will be considered as well.
The theory for trend filtering has been developed in a series of papers: Tibshirani [2014], Wang et al. [2016], Sadhanala and Tibshirani [2019], Sadhanala et al. [2017], Guntuboyina et al. [2020], and see Tibshirani [2020] for a very good overview and further references. In this course we will combine the approach of Dalalyan et al. [2017] with dual certificates as given in Candes and Plan [2011]. This allows us to obtain new results and ...

62J05 ; 62J99

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In this talk we look at polynomials having the property that all compositional iterates are irreducible, which we call dynamical irreducible. After surveying some previous results (mostly over finite fields), we will concentrate on the question of the dynamical irreducibility of integer polynomials being preserved in reduction modulo primes. More precisely, for a class of integer polynomials $f$, which in particular includes all quadratic polynomials, and also trinomials of some special form, we show that, under some natural conditions, he set of primes $p$ such that $f$ is dynamical irreducible modulo $p$ is of relative density zero. The proof of this result relies on a combination of analytic (the square sieve) and diophantine (finiteness of solutions to certain hyperelliptic equations) tools, which we will briefly describe.
In this talk we look at polynomials having the property that all compositional iterates are irreducible, which we call dynamical irreducible. After surveying some previous results (mostly over finite fields), we will concentrate on the question of the dynamical irreducibility of integer polynomials being preserved in reduction modulo primes. More precisely, for a class of integer polynomials $f$, which in particular includes all quadratic ...

11R09 ; 11R45 ; 11L40 ; 37P25

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We will discuss recent progress in analysis of uniform and ordinary Diophantine exponents $\hat\omega $ and $\omega$ for linear Diophantine approximation as well as some applications of the related methods. In particular, we give a new criterion for badly approximable vectors in $\mathbb{R}^{d}$ the behavior of the best approximation vectors in the sense of simultaneous approximation and in the sense of linear form. It turned out that compared to the one-dimensional case our criterion is rather unusual. We apply this criterion to the analysis of Dirichlet spectrum for simultaneous Diophantine approximation.
We will discuss recent progress in analysis of uniform and ordinary Diophantine exponents $\hat\omega $ and $\omega$ for linear Diophantine approximation as well as some applications of the related methods. In particular, we give a new criterion for badly approximable vectors in $\mathbb{R}^{d}$ the behavior of the best approximation vectors in the sense of simultaneous approximation and in the sense of linear form. It turned out that compared ...

11J13 ; 11J06 ; 11J70

Z