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A note on metric inhomogeneous Diophantine approximation

Part of: Diophantine approximation, transcendental number theory

Published online by Cambridge University Press:  09 April 2009

M. M. Dodson
M. M. Dodson
Affiliation:
Department of Mathematics University of YorkYork Yol 5DD, UK
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Abstract

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An inhomogeneous version of a general form of the Jarník-Besicovitch Theorem is proved.

MSC classification

Secondary: 11J83: Metric theory 11J20: Inhomogeneous linear forms
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 62 , Issue 2 , April 1997 , pp. 175 - 185
Copyright
Copyright © Australian Mathematical Society 1997

References

[1]Arnol'd, V. I., Geometrical methods in ordinary differential equations (Springer, Berlin, 1983) (translated by Szücs, J.).CrossRefGoogle Scholar
[2]Baker, A. and Schmidt, W. M., ‘Diophantine approximation and Hausdorff dimension’, Proc. London Math. Soc. 21 (1970), 1–11.CrossRefGoogle Scholar
[3]Besicovitch, A. S., ‘Sets of fractional dimensions (IV): On rational approximation to real numbers’, J. London Math. Soc. 9 (1934), 126–131.CrossRefGoogle Scholar
[4]Bovey, J. D. and Dodson, M. M., ‘The Hausdorff dimension of systems of linear forms’, Acta Arith. 45 (1984), 337–358.CrossRefGoogle Scholar
[5]Cassels, J. W. S., An introduction to diophantine approximation (Cambridge Univ. Press, London, 1957).Google Scholar
[6]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
[7]Dodson, M. M., ‘Geometric and probabilistic ideas in the metrical theory of Diophantine approximation’, Uspekhi Mat. Nauk 48 (1993), 77–106Google Scholar
(in Russian); also in Russian Math. Surveys 48 (1993), 73–102.CrossRefGoogle Scholar
[8]Dodson, M. M., Rynne, B. P. and Vickers, J. A. G., ‘Diophantine approximation and a lower bound for Hausdorff dimension’, Mathematika 37 (1990), 59–73.CrossRefGoogle Scholar
[9]Dodson, M. M., ‘The Hausdorff dimension of exceptional sets associated with normal forms’, J. London Math. Soc. 49 (1994), 614–624.CrossRefGoogle Scholar
[10]Dodson, M. M. and Vickers, J. A. G., ‘Exceptional sets in Kolmogorov-Arnol'd-Moser theory’, J. Phys. A. 19 (1986), 349–374.CrossRefGoogle Scholar
[11]Eggleston, H. G., ‘Sets of fractional dimension which occur in some problems in number theory’, Proc. London Math. Soc. 54 (1952), 42–93.CrossRefGoogle Scholar
[12]Jarník, V., ‘Diophantischen Approximationen und Hausdorffsches Mass’, Mat. Sb. 36 (1929), 371–382.Google Scholar
[13]Jarník, V., ‘Über die simultanen diophantischen Approximationen’, Math. Z. 33 (1931), 503–543.CrossRefGoogle Scholar
[14]Melián, M. V. and Pestana, D., ‘Geodesic excursions into cusps in finite volume hyperbolic manifolds’, Michigan Math. J. 40 (1993), 77–93.CrossRefGoogle Scholar
[15]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
[16]Schmidt, W., ‘A metrical theorem in Diophantine approximation’, Canad. J. Math. 12 (1960), 619–631.CrossRefGoogle Scholar
[17]Schmidt, W. M., ‘Metrical theorems on fractional parts of sequences’, Trans. Amer. Math. Soc. 10 (1964), 493–518.CrossRefGoogle Scholar
[18]Sprindzuk, V. G., Metric theory of Diophantine approximations (Wiley, New York, 1979) (translated by Silverman, R. A.).Google Scholar