Simultaneous approximation to a real number and its square
Let \(\xi\) be an irrational real number. From Dirichlet's box principle it follows that for any real number \(X\geq 1\), there exists \((x_{0},x_{1}, x_{2})\in\mathbb Z^{3}\) satisfying \[ 0<x_{0}\leq X,\quad | x_{0}\xi-x_{1}| \leq \varphi(X) \;\hbox{ and } \;| x_{0}\xi^{2}-x_{2}| \leq \varphi(X), \tag{1} \] where \(\varphi(X)=1/[\sqrt{X}]\). If \(\xi\) is algebraic of degree \(2\), the same is true with \(\varphi(X)=c/X\) where \(c>0\) depends only on \(\xi\). For \(\lambda>1/2\), it is known that the set \(E_{\lambda}\) of \(\xi\) which are not quadratic over \(\mathbb Q\) and for which these inequalities (1) have a solution for arbitrarily large values of \(X\) with \(\varphi(X)=X^{-\lambda}\) has Lebesgue measure zero (by a metrical result of V. G. Sprindzhuk) and contains no algebraic number (by the subspace theorem of W. M. Schmidt). No element in this set was known, and the general expectation was that this set should be empty. It was proved by \textit{H. Davenport} and \textit{W. M. Schmidt} [Acta Arith. 15, 393--416 (1969; Zbl 0186.08603)] that the set \(E_{\lambda}\) is empty for \(\lambda>\lambda_0=(-1+\sqrt{5})/2=0.618\dots\); more precisely for any irrational \(\xi\) which is not quadratic over \(\mathbb Q\) there is a constant \(c(\xi)\) such that the above inequalities (1) have no solution when \(X\) is sufficiently large and \(\varphi(X)=c(\xi)X^{-\lambda_{0}}\). In this note, the author shows that, surprisingly, this result of Davenport and Schmidt is optimal: he produces explicit examples of transcendental numbers \(\xi\) for which the inequalities (1) have a solution for arbitrarily large values of \(X\) with \(\varphi(X)=cX^{-\lambda_{0}}\) with a suitable constant \(c\). There is a close connection with the approximation of a real number by algebraic integers of degree \(\geq 3\), a problem which is also considered by the author in this paper. A detailed version of this work has been published by the author in [\textit{D. Roy}, Approximation to real numbers by cubic algebraic integers. I, Proc. Lond. Math. Soc. (3) 88, 42--62 (2004; Zbl 1035.11028); II, Ann. Math. (2) 158, 1081--1087 (2003; Zbl 1044.11061)].
scientific article; zbMATH DE number 1907813
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