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Hausdorff dimension, lower order and Khintchine’s theorem in metric Diophantine approximation. (English) Zbl 0749.11036

An analogue of Khintchine’s theorem in metric Diophantine approximation is proved in which the Lebesgue measure of the set \(W(\psi)\) of \(\psi\)- approximable points is replaced by Hausdorff dimension. In Khintchine’s theorem, \(W(\psi)\) has full or zero Lebesgue measure according as a volume sum involving the error function \(\psi\) diverges or converges. By contrast, in this variation of Khintchine’s theorem, the Hausdorff dimension of \(W(\psi)\) is expressed in terms of the lower order at infinity of the reciprocal \(1/\psi\) of \(\psi\), thus placing Hausdorff dimension on a footing comparable with Lebesgue measure and generalizing the classical results of Jarník and Besicovitch and their extensions. Neither theorem contains the other but they do overlap.

MSC:

11K55 Metric theory of other algorithms and expansions; measure and Hausdorff dimension
11K60 Diophantine approximation in probabilistic number theory
11J83 Metric theory
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