Quadratische Approximationen am Einheitskreis. (Quadratic approximations on the unit circle) (Q2266040)

Let A be the set of real numbers which are rationals or quadratic irrationals and H(\(\xi)\) the height of the algebraic number \(\xi\). Then the following approximation theorem for the unit circle U is proved: Let (\(\alpha\),\(\beta)\) be a point on U with \(\alpha\),\(\beta\) \(\not\in A\), then there exist infinitely many points (\(\lambda\),\(\mu)\) on U with \(\lambda\),\(\mu\in A\) such that \(| \alpha -\lambda | <C H(\lambda)^{-3/2}\), \(| \beta -\mu | <C H(\mu)^{-3/2}\) with \(C\geq 555\). Apart from the constant this result is best possible; the assertion becomes false if the exponent -3/2 is replaced by a smaller one. The proof is based on an important result of \textit{H. Davenport} and \textit{W. M. Schmidt} [Acta Arith. 13, 169-176 (1967; Zbl 0155.095)] on the approximation of real numbers \(\alpha\) \(\not\in A\) by numbers \(\xi\in A\).