Exponents of diophantine approximation in dimension 2 for numbers of Sturmian type

Poëls, Anthony

doi:10.1007/s00209-019-02280-2

Cited by 6 publications

(10 citation statements)

References 36 publications

(201 reference statements)

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“…Classical examples are extremal numbers as defined by Roy [27] and Sturmian continued fractions, see Bugeaud, Laurent [9], we omit definitions here. See also Poels [26]. For n = 2, k = 3 and ξ an extremal number, the identities from [27], [35] (10)…”

Section: Relations Between Exponents Of Simultaneous Approximationmentioning

confidence: 99%

“…First we notice that (25) enables the trivial condition w n (ξ) ≤ w n (ξ) for w n (ξ) > k (a slightly more restrictive bound follows from [25]), which we assume anyway for a non-trivial estimate. Assume ξ satisfies (26) w n (ξ) > w n−1 (ξ).…”

Section: Comparisonmentioning

confidence: 99%

“…Assume ξ is a real number and satisfies the inequality (26). Then If k = n we again obtain formula (23) as from Theorem 3.2, so then condition ( 26) is not required.…”

Section: Comparisonmentioning

confidence: 99%

“…Ordinary exponents may take the value +∞, whereas uniform exponents turn out to be always less than twice the trivial lower bounds in (3), see Remarks 1, 4 below for refinements. Only for N = 2 numbers satisfying λ N (ξ) > 1/N or w N (ξ) > N have been found, see Roy [27], [28], Fischler [16], Bugeaud, Laurent [9] and Poels [26].…”

Section: Introduction and Outlinementioning

confidence: 99%

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Going-up theorems for simultaneous Diophantine approximation

Schleischitz

2020

Preprint

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We establish several new inequalities linking classical exponents of Diophantine approximation associated to a real vector ξ = (ξ, ξ 2 , . . . , ξ N ), in various dimensions N . We thereby obtain variants, and partly refinements, of recent results of Badziahin and Bugeaud. We further implicitly recover inequalities of Bugeaud and Laurent as special cases, with new proofs. Similar estimates concerning general real vectors (not on the Veronese curve) with Q-linearly independent coordinates are addressed as well. Our method is based on Minkowski's Second Convex Body Theorem, applied in the framework of parametric geometry of numbers introduced by Schmidt and Summerer. We also frequently employ Mahler's Duality result on polar convex bodies.

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Section: Relations Between Exponents Of Simultaneous Approximationmentioning

confidence: 99%

Section: Comparisonmentioning

confidence: 99%

“…Assume ξ is a real number and satisfies the inequality (26). Then If k = n we again obtain formula (23) as from Theorem 3.2, so then condition ( 26) is not required.…”

Section: Comparisonmentioning

confidence: 99%

Section: Introduction and Outlinementioning

confidence: 99%

See 2 more Smart Citations

Going-up theorems for simultaneous Diophantine approximation

Schleischitz

2020

Preprint

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“…The numbers of Sturmian type introduced by A. Poëls in [6] include all Sturmian continued fractions mentioned above, and provide further examples of real numbers ξ for which the point (ξ, ξ 2 ) satisfies the hypotheses of our theorem. So, our measure (1.2) applies to these numbers as well.…”

Section: Introductionmentioning

confidence: 98%

A measure of transcendence for singular points on conics

Roy¹

2019

Journal De Théorie Des Nombres De Bordeaux

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A singular point on a plane conic defined over Q is a transcendental point of the curve which admits very good rational approximations, uniformly in terms of the height. Extremal numbers and Sturmian continued fractions are abscissa of such points on the parabola y = x 2 . In this paper we provide a measure of transcendence for singular points on conics defined over Q which, in these two cases, improves on the measure obtained by Adamczewski et Bugeaud. The main tool is a quantitative version of Schmidt subspace theorem due to Evertse.Résumé. Un point d'une conique définie sur Q est dit singulier s'il est transcendant et admet de très bonnes approximations rationnelles, uniformément en termes de la hauteur. Les nombres extrémaux et les fractions continues sturmiennes sont les abscisses de tels points sur la parabole y = x 2 . Nousétablissons ici une mesure de transcendance de points singuliers sur les coniques définies sur Q qui, dans ces deux cas, améliore la mesure obtenue précédemment par Adamczewski et Bugeaud. L'outil principal est une version quantitative du théorème du sous-espace de Schmidt dueà Evertse.

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Simultaneous rational approximation to successive powers of a real number

Poëls

Roy

2022

Trans. Amer. Math. Soc.

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We develop new tools leading, for each integer n ≥ 4 n\ge 4 , to a significantly improved upper bound for the uniform exponent of rational approximation λ ^ n ( ξ ) \widehat {\lambda }_n(\xi ) to successive powers 1 , ξ , … , ξ n 1,\xi ,\dots ,\xi ^n of a given real transcendental number ξ \xi . As an application, we obtain a refined lower bound for the exponent of approximation to ξ \xi by algebraic integers of degree at most n + 1 n+1 . The new lower bound is n / 2 + a n + 4 / 3 n/2+a\sqrt {n}+4/3 with a = ( 1 − log ⁡ ( 2 ) ) / 2 ≃ 0.153 a=(1-\log (2))/2\simeq 0.153 , instead of the current n / 2 + O ( 1 ) n/2+\mathcal {O}(1) .

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Exponents of diophantine approximation in dimension 2 for numbers of Sturmian type

Cited by 6 publications

References 36 publications

Going-up theorems for simultaneous Diophantine approximation

Going-up theorems for simultaneous Diophantine approximation

A measure of transcendence for singular points on conics

Simultaneous rational approximation to successive powers of a real number

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