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Rational approximation to real points on conics. (Approximation rationnelle de points réels sur les coniques.) (English. French summary) Zbl 1304.11064
Let $$\gamma:=(1+\sqrt{5})/2$$ denotes the golden ratio. Let $$n$$ be a positive integer and let $$\underline{\xi}=(\xi_1,\dots,\xi_n)\in\mathbb R^n.$$ The uniform exponent of approximation to $$\underline{\xi}$$ by rational points denoted $$\lambda(\underline{\xi})$$, is the supremum of all real numbers $$\lambda$$ for which the system of inequalities $$|x_0|\leq X,\;\max_{1\leq i\leq n}|x_0\xi_i-x_i|\leq X^{-\lambda}$$ admits a nonzero solution $$\mathbf x=(x_0,x_1\dots,x_n)\in \mathbb Z^{n+1}$$ for each sufficiently large real number $$X>1.$$
Let $$\mathcal C$$ be a closed algebraic subset of $$\mathbb R^n$$ of dimension $$1$$ defined over $$\mathbb Q$$, irreducible over $$\mathbb Q$$, and not contained in any proper affine linear subspace of $$\mathbb R^n$$ over $$\mathbb Q$$. Let $$\mathcal C^{\mathrm{li}}$$ denote the set of points $$\underline{\xi}=(\xi_1,\dots,\xi_n)\in\mathcal C$$ such that $$1,\xi_1,\dots,\xi_n$$ are linearly independent over $$\mathbb Q$$. Let $$\lambda(\mathcal C):=\sup\{\lambda(\underline{\xi});\underline{\xi}\in \mathcal C^{\mathrm{li}}\}\in\mathcal C^{\mathrm{li}}$$.
Historical context: H. Davenport and W. M. Schmidt [Acta Arith. 15, 393–416 (1969; Zbl 0186.08603)] have determined an upper bound $$\lambda_n$$ depending only on $$n$$, for $$\lambda(\xi,\xi^2,\dots,\xi^n),$$ where $$\xi$$ runs through all real numbers such that $$1,\xi,\dots,\xi^n$$ are linearly independent over $$\mathbb Q$$. For $$n=2$$ they have proved that $$\lambda_2=1/\gamma$$. See also [D. Roy, C. R., Math., Acad. Sci. Paris 336, No. 1, 1–6 (2003; Zbl 1038.11042)].
Results: The author proves the following result generalizing for all conics in $$\mathbb R^2$$ the result obtained in [Zbl 0186.08603] for parabolas:
Theorem 1. Let $$\mathcal C$$ be a closed algebraic subset of $$\mathbb R^2$$ of dimension $$1$$ and degree $$2$$. Suppose that $$\mathcal C$$ is defined over $$\mathbb Q$$ and irreducible over $$\mathbb Q$$. Then we have $$\lambda(\mathcal C)=1/\gamma$$. Moreover, the set of points $$\underline{\xi}\in \mathcal C^{\mathrm{li}}$$ with $$\lambda(\underline{\xi})=1/\gamma$$ is countably infinite.
The main result of the author is a stronger theorem generalizing Theorem 1 in projective settings. In particular, he proves also in this setting, that for curves $$\mathcal C$$ irreducible over $$\mathbb R$$ which contain at least one rational point, then the case reduces to the known case of parabolas.

##### MSC:
 11J13 Simultaneous homogeneous approximation, linear forms 14H50 Plane and space curves
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