Simultaneous approximation to a real number and its square. (Approximation simultanée d’un nombre et de son carré.) (French. Abridged English version) Zbl 1038.11042

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 H. Davenport and 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 [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)].


11J13 Simultaneous homogeneous approximation, linear forms
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[1] Allouche, J.-P.; Davison, J. L.; Queffélec, M.; Zamboni, L. Q., Transcendence of Sturmian or morphic continued fractions, J. Number Theory, 91, 39-66 (2001) · Zbl 0998.11036
[2] Bugeaud, Y.; Teulié, O., Approximation d’un nombre réel par des nombres algébriques de degré donné, Acta Arith., 93, 77-86 (2000) · Zbl 0948.11029
[3] Cassels, J. W.S., An Introduction to Diophantine Approximation (1957), Cambridge University Press · Zbl 0077.04801
[4] Davenport, H.; Schmidt, W. M., Approximation to real numbers by algebraic integers, Acta Arith., 15, 393-416 (1969), (=H. Davenport, Coll. Works, Vol. II, No. 183) · Zbl 0186.08603
[5] de Luca, A., A combinatorial property of the Fibonacci words, Inform. Process. Lett., 12, 193-195 (1981) · Zbl 0468.20049
[6] Lothaire, M., Combinatorics on Words. Combinatorics on Words, Encyclopedia of Mathematics and Its Applications, 17 (1983), Addison-Wesley · Zbl 0514.20045
[7] Roy, D., Approximation to real numbers by cubic algebraic integers I, Proc. London Math. Soc., (to appear)
[8] Schmidt, W. M., Diophantine Approximation, Lecture Notes in Math., Vol. 785 (1980), Springer-Verlag · Zbl 0421.10019
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