On the number of good rational approximations to algebraic numbers. (English) Zbl 0675.10023

Let \(\xi\) be a real number, and \(\delta\) a positive number. Define \(L=\log (1+\delta)\). Using a “gap principle”, the authors prove that the number of rational numbers x/y\(\in {\mathbb{Q}}\) with denominator \(0<y<B\) satisfying \(| \xi -x/y| <y^{-2-\delta}\) is at most \((1/L)\log \log B+c(\delta).\) Moreover, they construct a real transcendental number \(\xi\) which shows that this estimate is best possible as far as the dependence on B is concerned.
They also consider the approximation to an algebraic number \(\xi\) of degree r and height H: the number of approximations x/y\(\in {\mathbb{Q}}\) (with denominator \(y>0)\) satisfying \(| \xi -x/y| <y^{-2- \delta}\) is at most \((1/L) \log \log H+c'(\delta,r).\) The authors derive this statement by using the preceding result for the approximations with a “small” denominator, and using a result of E. Bombieri and A. J. van der Poorten [J. Aust. Math. Soc., Ser. A 45, No.2, 233-248 (1988; Zbl 0664.10017)] for “large” denominators. Once more, the first summand in the estimate is the best possible one, and the proof of this fact involves a result of E. Wirsing [J. Reine Angew. Math. 206, 67-77 (1961; Zbl 0097.035)].
Reviewer: M.Waldschmidt


11J68 Approximation to algebraic numbers
11J17 Approximation by numbers from a fixed field
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