Geodesic
A note on the theorem of Jarník-Besicovitch
Glasgow mathematical journal, Tome 39 (1997) no. 2, pp. 233-236

Voir la notice de l'article provenant de la source Cambridge University Press

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. 
Dickinson, H. A note on the theorem of Jarník-Besicovitch. Glasgow mathematical journal, Tome 39 (1997) no. 2, pp. 233-236. doi: 10.1017/S0017089500032134
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[1] 1.Besicovitch, A. S., Sets of fractional dimension (IV): On rational approximation to real numbers, J. London Math. Soc. 9 (1934), 126–131. Google Scholar | DOI

[2] 2.Bovey, J. and Dodson, M. M., The Hausdorff dimension of systems of linear forms, Acta Arith. 45 (1986), 337–358. Google Scholar | DOI

[3] 3.Borosh, I. and Fraenkel, A. S., A generalization of Jarnik's theorem on Diophantine approximations, Indag. Math. 34 (1972), 193–201. Google Scholar | DOI

[4] 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] 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] 6.Dodson, M. M., A note on metric inhomogeneous Diophantine approximation, J. Austral. Math. Soc., to appear. Google Scholar

[7] 7.Dodson, M. M., Rynne, B. P. and Vickers, J., Diophantine approximation and a lower bound for Hausdorff dimension, Mathematika 37 (1990), 59–73. Google Scholar | DOI

[8] 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] 9.Forrest, A. H., The limit points of certain classical exponential series and other continuous cocycles, Preprint. Google Scholar

[10] 10.Groshev, A. V., Un Thé'orème sur les systèmes des formes linéaires, Dokl. Akad. Nauk SSSR 19 (1938), 151–152. Google Scholar

[11] 11.Hayman, W. K., Multivalent functions (2nd EditionCambridge University Press, 1994). Google Scholar | DOI

[12] 12.Hinokuma, T. and Shiga, H., A remark on a Theorem ofJarník, Ryukyu Math. J. 5 (1992), 1–6. Google Scholar

[13] 13.Hinokuma, T. and Shiga, H., Hansdorff dimension of sets arising in Diophantine approximation, Kodai Math. J. 19 (1996) No. 3, 365–377. Google Scholar | DOI

[14] 14.Jarník, V., Diophantische Approximationen und Hausdorffsches Mass, Mat. Sb. 36 (1929), 371–382. Google Scholar

[15] 15.Jarník, V., Über die simultanen diophantischen Approximationen, Math. Z. 33 (1931), 505–543. Google Scholar | DOI

[16] 16.Khintchine, A., Einige Sätze über Kettenbriiche, mit Anwendungen auf die Theorie der Diophantischen Approximationen, Math. Ann. 92 (1924), 115–125. Google Scholar | DOI

[17] 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. Google Scholar | DOI

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