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SIMULTANEOUS DIOPHANTINE APPROXIMATION ON POLYNOMIAL CURVES

Part of: Diophantine approximation, transcendental number theory

Published online by Cambridge University Press:  10 December 2009

Natalia Budarina ,
Detta Dickinson  and
Jason Levesley
Natalia Budarina
Affiliation:
Department of Mathematics, Logic House, NUI Maynooth, Co Kildare, Republic of Ireland
Detta Dickinson
Affiliation:
Department of Mathematics, Logic House, NUI Maynooth, Co Kildare, Republic of Ireland (email: [email protected])
Jason Levesley
Affiliation:
Department of Mathematics, University of York, Heslington, York YO10 5DD, U.K.
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Abstract

The Hausdorff dimension and measure of the set of simultaneously ψ-approximable points lying on integer polynomial curves is obtained for sufficiently small error functions.

MSC classification

Secondary: 11J13: Simultaneous homogeneous approximation, linear forms 11J83: Metric theory 11J54: Small fractional parts of polynomials and generalizations
Type
Research Article
Information
Mathematika , Volume 56 , Issue 1 , January 2010 , pp. 77 - 85
Copyright
Copyright © University College London 2010

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References

[1]Baker, R. C., Dirichlet’s theorem on Diophantine approximation. Math. Proc. Cambridge Philos. Soc. 83 (1978), 37–59.CrossRefGoogle Scholar
[2]Beresnevich, V., Distribution of rational points near a parabola. Dokl. Nats. Akad. Nauk Belarusi 45 (2001), 21–23 (in Russian).Google Scholar
[3]Beresnevich, V. V., A Groshev type theorem for convergence on manifolds. Acta Math. Hungar. 94 (2002), 99–130.CrossRefGoogle Scholar
[4]Beresnevich, V., Bernik, V., Kleinbock, D. and Margulis, G., Metric Diophantine approximation, the Khintchine–Groshev theorem for non-degenerate manifolds. Mosc. Math. J. 2(2) (2002), 203–225.CrossRefGoogle Scholar
[5]Beresnevich, V., Dickinson, D. and Velani, S., Diophantine approximation on planar curves and the distribution of rational points. Ann. of Math. (2) 166(2) (2007), 367–426.CrossRefGoogle Scholar
[6]Bernik, V. I., An application of Hausdorff dimension in the theory of Diophantine approximation. Acta Arith. 42 (1983), 219–253 (in Russian).Google Scholar
[7]Bernik, V. I. and Dodson, M. M., Metric Diophantine Approximation on Manifolds and Hausdorff Dimension (Cambridge Tracts in Mathematics 137), Cambridge University Press (Cambridge, 1999).CrossRefGoogle Scholar
[8]Bernik, V., Kleinbock, D. and Margulis, G. A., Khintchine-type theorems on manifolds: the convergence case for standard and multiplicative versions. Int. Math. Res. Not. 9 (2001), 453–486.CrossRefGoogle Scholar
[9]Budarina, N. and Dickinson, D., Simultaneous Diophantine approximation on surfaces defined by simple polynomial expressions. Math. Proc. R. Ir. Acad. (2009) (submitted).CrossRefGoogle Scholar
[10]Bugeaud, Y. and Laurent, M., Exponents of Diophantine approximation. In Diophantine Geometry (Centro di Ricerca Matematica Ennio De Giorgi (CRM) Series 4) (2007) 101–121.Google Scholar
[11]Dickinson, D., Ideas and results from the theory of Diophantine approximation. Conference Proceedings: Diophantine Phenomena in Differential Equations and Dynamical Systems (RIMS Kyoto 2004).Google Scholar
[12]Dickinson, H. and Dodson, M. M., Extremal manifolds and Hausdorff dimension. Duke Math. J. 101(2) (2000), 271–281.CrossRefGoogle Scholar
[13]Dickinson, H. and Dodson, M. M., Simultaneous Diophantine approximation on the circle and Hausdorff dimension. Math. Proc. Cambridge Philos. Soc. 130 (2001), 515–522.CrossRefGoogle Scholar
[14]Dickinson, D. and Velani, S., Hausdorff measure and linear forms. J. Reine Angew. Math. 490 (1997), 1–36.Google Scholar
[15]Dodson, M. M., Rynne, B. P. and Vickers, J. A. G., Metric Diophantine approximation and Hausdorff dimension on manifolds. Math. Proc. Cambridge Philos. Soc. 105 (1989), 547–558.CrossRefGoogle Scholar
[16]Dodson, M. M., Rynne, B. P. and Vickers, J. A. G., Khintchine-type theorems on manifolds. Acta Arith. 57 (1991), 115–130.CrossRefGoogle Scholar
[17]Dodson, M. M., Rynne, B. P. and Vickers, J. A. G., Simultaneous Diophantine approximation and asymptotic formulae on manifolds. J. Number Theory 58 (1996), 298–316.CrossRefGoogle Scholar
[18]Drutu, C., Diophantine approximation on rational quadrics. Math. Ann. 333 (2005), 405–469.CrossRefGoogle Scholar
[19]Falconer, K., Fractal Geometry, Wiley (New York, 1989).Google Scholar
[20]Jarník, V., Über die simultanen Diophantischen Aproximationen. Math. Z. 33 (1931), 505–543.CrossRefGoogle Scholar
[21]Kleinbock, D. Y., Extremal subspaces and their sub-manifolds. Geom. Funct. Anal. 13(2) (2003), 437–466.CrossRefGoogle Scholar
[22]Kleinbock, D. Y. and Margulis, G. A., Flows on homogeneous spaces and Diophantine approximation on manifolds. Ann. of Math. (2) 148(2) (1998), 339–360.CrossRefGoogle Scholar