A variational principle in the parametric geometry of numbers, with applications to metric Diophantine approximation

Das, Tushar; Fishman, Lior; Simmons, David; Urbański, Mariusz

doi:10.1016/j.crma.2017.07.007

Cited by 24 publications

(64 citation statements)

References 16 publications

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“…. When min{m, n} 2, important contributions towards Problem 2 are announced without proofs by T. Das, L. Fishman, D. Simmons and M. Urbański, see [13]. Tedious calculation shows that the lower bound given in Theorem 3 agrees with the formula given in [13, theorem 1•9].…”

Section: Theorem 1 For Any Real Numbermentioning

confidence: 73%

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Hausdorff dimension and uniform exponents in dimension two

Bugeaud¹,

Cheung²,

Chevallier³

2018

Math. Proc. Camb. Phil. Soc.

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In this paper we prove the Hausdorff dimension of the set of (nondegenerate) singular two-dimensional vectors with uniform exponent µ ∈ (1/2, 1) is 2(1 − µ) when µ ≥ √ 2/2, whereas for µ < √ 2/2 it is greater than 2(1 − µ) and at most (3 − 2µ)(1 − µ)/(1 + µ + µ 2 ). We also establish that this dimension tends to 4/3 (which is the dimension of the set of singular two-dimensional vectors) when µ tends to 1/2. These results improve upon previous estimates of R. Baker, joint work of the first author with M. Laurent, and unpublished work of M. Laurent. We also prove a lower bound on the packing dimension that is strictly greater than the Hausdorff dimension for µ ≥ 0.565 . . . . Definition 1. Let m, n be positive integers and A a real n × m matrix. The matrix A is badly approximable if there exists a positive constant c such that the system of inequalitieshas no solution x in Z m for any X ≥ 1.Definition 2. Let m, n be positive integers and A a real n × m matrix. We say that Dirichlet's Theorem can be improved for the matrix A if there exists a positive constant c < 1 such that the system of inequalities (1.1) has a solution x in Z m for any sufficiently large X.If the subgroup G = AZ m + Z n of R n generated by the m rows of the matrix t A (here and below, t M denotes the transpose of a matrix M) together with Z n has rank strictly less than m + n, then there exists x in Z m with |x| ∞ arbitrarily large, such that Ax = 0 and, consequently, for any real number w and any sufficiently large X > 1, the system of inequalitieshas a solution x in Z m . In several of the questions considered below, we have to exclude this degenerate situation, thus we are led to introduce the set M * n,m (R) of n × m matrices for which the associated subgroup G has rank m + n.When m = n = 1, that is, when A = (ξ) for some irrational real number ξ, it is not difficult to show that Dirichlet's Theorem can be improved if, and only if, ξ is badly approximable (or, equivalently, ξ has bounded partial quotients in its continued fraction expansion); see [19] and [13] for a precise statement. Furthermore, by using the theory of continued fractions, one can prove that, for any irrational real number ξ, there are arbitrarily large integers X such that the system of inequalities xξ|| ≤ 1 2X and 0 < x ≤ X (1.2) has no integer solutions; see Proposition 2.2.4 of [5].Since the set of badly approximable numbers has Lebesgue measure zero and Hausdorff dimension 1, this implies that the set of 1 × 1 matrices A for which Dirichlet's Theorem can be improved has Lebesgue measure zero and Hausdorff dimension 1. The latter assertion has been extended as follows.Theorem A. For any positive integers m, n, the set of real n×m matrices for which Dirichlet's Theorem can be improved has mn-dimensional Lebesgue measure zero and Hausdorff dimension mn.The first assertion of Theorem A has been established by Davenport and Schmidt [14] when min{m, n} = 1. According to Kleinbock and Weiss [21], their proof can be generalized to n × m matrices. Actually, a more general re...

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Section: Theorem 1 For Any Real Numbermentioning

confidence: 73%

“…Kadyrov et al [20] established that this dimension is bounded from above by mn(m + n − 1)/(m + n) and it is conjectured that there is in fact equality. In [13], Das et al announced a proof of this conjecture as a consequence of a "variational principle".…”

Section: Theorem B the Hausdorff Dimension Of The Set Of Singular Twmentioning

confidence: 98%

Hausdorff dimension and uniform exponents in dimension two

Bugeaud¹,

Cheung²,

Chevallier³

2018

Math. Proc. Camb. Phil. Soc.

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“…In[2,3], Das, Fishman, Simmons and Urbański introduce a variational principle in parametric geometry of numbers that extends Theorem 2. They both extend to the case of approximation to a matrix θ, and provide a quantitative result.…”

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confidence: 99%

An Optimal Bound for the Ratio Between Ordinary and Uniform Exponents of Diophantine Approximation

Marnat

Moshchevitin

2020

Mathematika

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We provide a lower bound for the ratio between the ordinary and uniform exponents of both simultaneous Diophantine approximation to n real numbers and Diophantine approximation for one linear form in n variables. This question was first considered in the 1950s by Jarník who solved the problem for two real numbers and established certain bounds in higher dimension. Recently different authors reconsidered the question, solving the problem in dimension three with different methods. Considering a new concept of parametric geometry of numbers, Schmidt and Summerer conjectured that the optimal lower bound is reached at regular systems. It follows from a remarkable result of Roy that this lower bound is then optimal. In the present paper, we give a proof of this conjecture by Schmidt and Summerer.

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“…, α n , 1} linearly independent over Q exist if n > 1. It has recently been shown that when n > 1 the set of singular α ∈ R n has Hausdorff dimension n 2 n+1 (see [3], [4], and [6]). The Minkowski chain also gives a criterion for L α to be singular.…”

Section: Applications To Diophantine Approximationmentioning

confidence: 99%

The Minkowski chain and Diophantine approximation

Andersen¹,

Duke²

2020

Journal De Théorie Des Nombres De Bordeaux

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L'accès aux articles de la revue « Journal de Théorie des Nombres de Bordeaux » (http://jtnb.centre-mersenne.org/), implique l'accord avec les conditions générales d'utilisation (http://jtnb. centre-mersenne.org/legal/). Toute reproduction en tout ou partie de cet article sous quelque forme que ce soit pour tout usage autre que l'utilisation à fin strictement personnelle du copiste est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.centre-mersenne.org/ Journal de Théorie des Nombres de Bordeaux 32 (2020), 503-523 The Minkowski chain and Diophantine approximation par Nickolas ANDERSEN et William DUKE Résumé. La chaîne de Hurwitz donne une suite de paires d'approximations de Farey d'un nombre réel irrationnel. Minkowski a donné un critère d'algébraicité d'un nombre en utilisant une certaine généralisation de la chaîne de Hurwitz. Nous appliquons cette généralisation (la chaîne de Minkowski) pour donner des critères pour qu'une forme linéaire réelle soit mal approchable ou singulière. Les preuves reposent sur des propriétés des minima successifs et des bases réduites de réseaux.

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A variational principle in the parametric geometry of numbers, with applications to metric Diophantine approximation

Cited by 24 publications

References 16 publications

Hausdorff dimension and uniform exponents in dimension two

Hausdorff dimension and uniform exponents in dimension two

An Optimal Bound for the Ratio Between Ordinary and Uniform Exponents of Diophantine Approximation

The Minkowski chain and Diophantine approximation

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