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A note on the theorem of Jarník-Besicovitch

Published online by Cambridge University Press:  18 May 2009

H. Dickinson
H. Dickinson
Affiliation:
Department of Mathematics, University of York, Heslington, York Y01 5DD, England
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This note draws together and extends two recent results on Diophantine approximation and Hausdorff dimension. The first, by Hinokuma and Shiga [12], considers the oscillating error function | sinq|/qτ rather than the strictly decreasing function qτ of Jarnik's theorem. The second is Rynne's extension [17] to systems of linear forms of Borosh and Fraenkel's paper [3] on restricted Diophantine approximation with real numbers. Rynne's result will be extended to a class of general error functions and applied to obtain a more general form of [12] in which the error function is any positive function.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 39 , Issue 2 , May 1997 , pp. 233 - 236
Copyright
Copyright © Glasgow Mathematical Journal Trust 1997

References

REFERENCES

1.Besicovitch, A. S., Sets of fractional dimension (IV): On rational approximation to real numbers, J. London Math. Soc. 9 (1934), 126–131.CrossRefGoogle Scholar
2.Bovey, J. and Dodson, M. M., The Hausdorff dimension of systems of linear forms, Acta Arith. 45 (1986), 337–358.CrossRefGoogle Scholar
3.Borosh, I. and Fraenkel, A. S., A generalization of Jarnik's theorem on Diophantine approximations, Indag. Math. 34 (1972), 193–201.CrossRefGoogle Scholar
4.Dickinson, H. and Velani, S., Hausdorff dimension and linear forms, Preprint Mathematica Gottingensis, Heft 23 (1995). (J. Reine Angew. Math., to appear)Google Scholar
5.Dodson, M. M., Hausdorff dimension, lower order and Khintchine's theorem in metric Diophantine approximation, J. Reine Angew. Math. 432 (1992), 69–76.Google Scholar
6.Dodson, M. M., A note on metric inhomogeneous Diophantine approximation, J. Austral. Math. Soc., to appear.Google Scholar
7.Dodson, M. M., Rynne, B. P. and Vickers, J., Diophantine approximation and a lower bound for Hausdorff dimension, Mathematika 37 (1990), 59–73.CrossRefGoogle Scholar
8.Eggleston, H. G., Sets of fractional dimensions which occur in some problems of number theory, Proc. London Math. Soc. 54 (1951), 42–93.Google Scholar
9.Forrest, A. H., The limit points of certain classical exponential series and other continuous cocycles, Preprint.Google Scholar
10.Groshev, A. V., Un Thé'orème sur les systèmes des formes linéaires, Dokl. Akad. Nauk SSSR 19 (1938), 151–152.Google Scholar
11.Hayman, W. K., Multivalent functions (2nd EditionCambridge University Press, 1994).CrossRefGoogle Scholar
12.Hinokuma, T. and Shiga, H., A remark on a Theorem ofJarník, Ryukyu Math. J. 5 (1992), 1–6.Google Scholar
13.Hinokuma, T. and Shiga, H., Hansdorff dimension of sets arising in Diophantine approximation, Kodai Math. J. 19 (1996) No. 3, 365–377.CrossRefGoogle Scholar
14.Jarník, V., Diophantische Approximationen und Hausdorffsches Mass, Mat. Sb. 36 (1929), 371–382.Google Scholar
15.Jarník, V., Über die simultanen diophantischen Approximationen, Math. Z. 33 (1931), 505–543.CrossRefGoogle Scholar
16.Khintchine, A., Einige Sätze über Kettenbriiche, mit Anwendungen auf die Theorie der Diophantischen Approximationen, Math. Ann. 92 (1924), 115–125.CrossRefGoogle Scholar
17.Rynne, B. P., The Hausdorff dimension of certain sets arising from Diophantine approximation by restricted sequences of integer vectors, Acta Arith., 61 (1992), 69–81.CrossRefGoogle Scholar