## 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

### MSC:

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

### Citations:

Zbl 0664.10017; Zbl 0097.035
Full Text:

### References:

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