A measure of transcendence for singular points on conics
Damien Roy1
1 Département de Mathématiques Université d’Ottawa 150 Louis Pasteur Ottawa, Ontario K1N 6N5, Canada
Journal de théorie des nombres de Bordeaux, Volume 31 (2019) no. 2, pp. 361-369.

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 and Bugeaud. The main tool is a quantitative version of Schmidt subspace theorem due to Evertse.

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 . Nous établissons 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 due à Evertse.

Received:
Accepted:
Published online:
DOI: 10.5802/jtnb.1085
Classification: 11J82, 11J13, 11J87
Mots-clés : Sturmian continued fractions, extremal numbers, transcendental numbers, measure of transcendence, uniform approximation, quantitative subspace theorem, minimal points, conics.

Damien Roy 1

1 Département de Mathématiques Université d’Ottawa 150 Louis Pasteur Ottawa, Ontario K1N 6N5, Canada
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Damien Roy. A measure of transcendence for singular points on conics. Journal de théorie des nombres de Bordeaux, Volume 31 (2019) no. 2, pp. 361-369. doi : 10.5802/jtnb.1085. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1085/

[1] Boris Adamczewski; Yann Bugeaud Mesures de transcendance et aspects quantitatifs de la méthode de Thue–Siegel–Roth–Schmidt, Proc. Lond. Math. Soc., Volume 101 (2010) no. 1, pp. 1-26 | DOI | Zbl

[2] Boris Adamczewski; Yann Bugeaud Transcendence measures for continued fractions involving repetitive or symmetric patterns, J. Eur. Math. Soc., Volume 12 (2010) no. 4, pp. 883-914 | DOI | MR | Zbl

[3] Yann Bugeaud; Michel Laurent Exponents of Diophantine approximation and Sturmian continued fractions, Ann. Inst. Fourier, Volume 55 (2005) no. 3, pp. 773-804 | DOI | MR | Zbl

[4] Harold Davenport; Wolfgang M. Schmidt Approximation to real numbers by quadratic irrationals, Acta Arith., Volume 13 (1967), pp. 169-176 | DOI | MR | Zbl

[5] Harold Davenport; Wolfgang M. Schmidt Approximation to real numbers by algebraic integers, Acta Arith., Volume 15 (1969), pp. 393-416 | DOI | MR | Zbl

[6] Jan-Hendrik Evertse An improvement of the quantitative subspace theorem, Compos. Math., Volume 101 (1996) no. 3, pp. 225-311 | MR | Zbl

[7] Anthony Poëls Exponents of diophantine approximation in dimension 2 for numbers of Sturmian type (2017) (to appear in Math. Z., https://arxiv.org/abs/1711.07896) | Zbl

[8] Damien Roy Approximation simultanée d’un nombre et de son carré, C. R. Math. Acad. Sci. Paris, Volume 336 (2003) no. 1, pp. 1-6 | MR | Zbl

[9] Damien Roy Approximation to real numbers by cubic algebraic integers I, Proc. Lond. Math. Soc., Volume 88 (2004) no. 1, pp. 42-62 | MR | Zbl

[10] Damien Roy Rational approximation to real points on conics, Ann. Inst. Fourier, Volume 63 (2013) no. 6, pp. 2331-2348 | MR | Zbl

[11] Wolfgang M. Schmidt Diophantine approximation, Lecture Notes in Mathematics, 785, Springer, 1980 | Zbl

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