Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-23T03:29:42.328Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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