Dodson, M. M. Hausdorff dimension, lower order and Khintchine’s theorem in metric Diophantine approximation. (English) Zbl 0749.11036 J. Reine Angew. Math. 432, 69-76 (1992). 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. Reviewer: M.M.Dodson (Heslington) Cited in 1 ReviewCited in 20 Documents MSC: 11K55 Metric theory of other algorithms and expansions; measure and Hausdorff dimension 11K60 Diophantine approximation in probabilistic number theory 11J83 Metric theory Keywords:analogue of Khintchine’s theorem; metric Diophantine approximation; Hausdorff dimension PDFBibTeX XMLCite \textit{M. M. Dodson}, J. Reine Angew. Math. 432, 69--76 (1992; Zbl 0749.11036) Full Text: DOI Crelle EuDML