# 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

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topRoy, 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 -

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