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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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References:
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