Rational approximation to real points on conics

Damien Roy[1]

  • [1] Université d’Ottawa Département de Mathématiques 585 King Edward Ottawa, Ontario K1N 6N5 (Canada)

Annales de l’institut Fourier (2013)

  • Volume: 63, Issue: 6, page 2331-2348
  • ISSN: 0373-0956

Abstract

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A point ( ξ 1 , ξ 2 ) with coordinates in a subfield of ℝ of transcendence degree one over ℚ , with 1 , ξ 1 , ξ 2 linearly independent over ℚ , may have a uniform exponent of approximation by elements of ℚ 2 that is strictly larger than the lower bound 1 / 2 given by Dirichlet’s box principle. This appeared as a surprise, in connection to work of Davenport and Schmidt, for points of the parabola { ( ξ , ξ 2 ) ; ξ ∈ ℝ } . The goal of this paper is to show that this phenomenon extends to all real conics defined over ℚ , and that the largest exponent of approximation achieved by points of these curves satisfying the above condition of linear independence is always the same, independently of the curve, namely 1 / γ ≅ 0 . 618 where γ denotes the golden ratio.

How to cite

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Roy, Damien. "Rational approximation to real points on conics." Annales de l’institut Fourier 63.6 (2013): 2331-2348. <http://eudml.org/doc/275520>.

@article{Roy2013,
abstract = {A point $(\xi _1,\xi _2)$ with coordinates in a subfield of $\mathbb\{R\}$ of transcendence degree one over $\{\mathbb\{Q\}\}$, with $1,\xi _1,\xi _2$ linearly independent over $\{\mathbb\{Q\}\}$, may have a uniform exponent of approximation by elements of $\{\mathbb\{Q\}\}^2$ that is strictly larger than the lower bound $1/2$ given by Dirichlet’s box principle. This appeared as a surprise, in connection to work of Davenport and Schmidt, for points of the parabola $\lbrace (\xi ,\xi ^2) \,;\, \xi \in \{\mathbb\{R\}\}\rbrace $. The goal of this paper is to show that this phenomenon extends to all real conics defined over $\{\mathbb\{Q\}\}$, and that the largest exponent of approximation achieved by points of these curves satisfying the above condition of linear independence is always the same, independently of the curve, namely $1/\gamma \cong 0.618$ where $\gamma $ denotes the golden ratio.},
affiliation = {Université d’Ottawa Département de Mathématiques 585 King Edward Ottawa, Ontario K1N 6N5 (Canada)},
author = {Roy, Damien},
journal = {Annales de l’institut Fourier},
keywords = {algebraic curves; conics; real points; approximation by rational points; exponent of approximation; simultaneous approximation; simultaneous approximations},
language = {eng},
number = {6},
pages = {2331-2348},
publisher = {Association des Annales de l’institut Fourier},
title = {Rational approximation to real points on conics},
url = {http://eudml.org/doc/275520},
volume = {63},
year = {2013},
}

TY - JOUR
AU - Roy, Damien
TI - Rational approximation to real points on conics
JO - Annales de l’institut Fourier
PY - 2013
PB - Association des Annales de l’institut Fourier
VL - 63
IS - 6
SP - 2331
EP - 2348
AB - A point $(\xi _1,\xi _2)$ with coordinates in a subfield of $\mathbb{R}$ of transcendence degree one over ${\mathbb{Q}}$, with $1,\xi _1,\xi _2$ linearly independent over ${\mathbb{Q}}$, may have a uniform exponent of approximation by elements of ${\mathbb{Q}}^2$ that is strictly larger than the lower bound $1/2$ given by Dirichlet’s box principle. This appeared as a surprise, in connection to work of Davenport and Schmidt, for points of the parabola $\lbrace (\xi ,\xi ^2) \,;\, \xi \in {\mathbb{R}}\rbrace $. The goal of this paper is to show that this phenomenon extends to all real conics defined over ${\mathbb{Q}}$, and that the largest exponent of approximation achieved by points of these curves satisfying the above condition of linear independence is always the same, independently of the curve, namely $1/\gamma \cong 0.618$ where $\gamma $ denotes the golden ratio.
LA - eng
KW - algebraic curves; conics; real points; approximation by rational points; exponent of approximation; simultaneous approximation; simultaneous approximations
UR - http://eudml.org/doc/275520
ER -

References

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  2. Yann Bugeaud, Michel Laurent, Exponents of Diophantine approximation and Sturmian continued fractions, Ann. Inst. Fourier (Grenoble) 55 (2005), 773-804 Zbl1155.11333MR2149403
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  8. Damien Roy, Approximation to real numbers by cubic algebraic integers. II, Ann. of Math. (2) 158 (2003), 1081-1087 Zbl1044.11061MR2031862
  9. Damien Roy, Approximation to real numbers by cubic algebraic integers. I, Proc. London Math. Soc. (3) 88 (2004), 42-62 Zbl1035.11028MR2018957
  10. Damien Roy, On two exponents of approximation related to a real number and its square, Canad. J. Math. 59 (2007), 211-224 Zbl1115.11036MR2289424
  11. Damien Roy, On simultaneous rational approximations to a real number, its square, and its cube, Acta Arith. 133 (2008), 185-197 Zbl1228.11100MR2417464
  12. Wolfgang M. Schmidt, Diophantine approximation, 785 (1980), Springer, Berlin Zbl0421.10019MR568710
  13. Dmitrij Zelo, Simultaneous approximation to real and p-adic numbers, (2009), ProQuest LLC, Ann Arbor, MI Zbl1241.11071MR2736753

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