## 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:

 [1] E. Bombieri and A. J. van der Poorten, Some quantitative results related to Roth’s theorem, J. Austral. Math. Soc. Ser. A 45 (1988), no. 2, 233 – 248. · Zbl 0664.10017 [2] H. Davenport and K. F. Roth, Rational approximations to algebraic numbers, Mathematika 2 (1955), 160 – 167. · Zbl 0066.29302 [3] G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, Oxford, at the Clarendon Press, 1954. 3rd ed. · Zbl 0058.03301 [4] H. Luckhardt, Herbrand-Analysen zweier Beweise des Satzes von Roth: Polynomiale Anzahlschranken, J. Symbolic Logic 54 (1989), no. 1, 234 – 263 (German, with English summary). · Zbl 0669.03024 [5] K. Mahler, An application of Jensen’s formula to polynomials, Mathematika 7 (1960), 98 – 100. · Zbl 0099.25003 [6] Wolfgang M. Schmidt, On the number of good simultaneous approximations to algebraic numbers, International Symposium in Memory of Hua Loo Keng, Vol. I (Beijing, 1988) Springer, Berlin, 1991, pp. 249 – 264. · Zbl 0813.11039 [7] Eduard Wirsing, Approximation mit algebraischen Zahlen beschränkten Grades, J. Reine Angew. Math. 206 (1961), 67 – 77 (German). · Zbl 0097.03503
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