Hostname: page-component-76fb5796d-9pm4c Total loading time: 0 Render date: 2024-04-25T14:44:19.823Z Has data issue: false hasContentIssue false
  • English
  • Français

A Jarník type theorem for planar curves: everything about the parabola

Published online by Cambridge University Press:  01 April 2015

MUMTAZ HUSSAIN
MUMTAZ HUSSAIN*
Affiliation:
School of Mathematical and Physical Sciences, The University of Newcastle, Callaghan, NSW 2308, Australia. e-mail: mumtaz.hussain@newcastle.edu.au
Get access Rights & Permissions [Opens in a new window]

Abstract

The well-known theorems of Khintchine and Jarník in metric Diophantine approximation provide a comprehensive description of the measure theoretic properties of real numbers approximable by rational numbers with a given error. Various generalisations of these fundamental results have been obtained for other settings, in particular, for curves and more generally manifolds. In this paper we develop the theory for planar curves by completing the theory in the case of parabola. This represents the first comprehensive study of its kind in the theory of Diophantine approximation on manifolds.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 159 , Issue 1 , July 2015 , pp. 47 - 60
Copyright
Copyright © Cambridge Philosophical Society 2015 

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

REFERENCES

[1] Badziahin, D. Inhomogeneous Diophantine approximation on curves and Hausdorff dimension. Adv. Math. 223 (2010), 329–351.Google Scholar
[2] Badziahin, D., Beresnevich, V. and Velani, S. Inhomogeneous theory of dual Diophantine approximation on manifolds. Adv. Math. 232 (2013), 1–35.Google Scholar
[3] Baker, A. and Schmidt, W. M. Diophantine approximation and Hausdorff dimension. Proc. London Math. Soc. (3) 21 (1970), 1–11.Google Scholar
[4] Baker, R. C. Dirichlet's theorem on Diophantine approximation. Math. Proc. Camb. Phil. Soc. 83 (1) (1978), 37–59.Google Scholar
[5] Beresnevich, V. Effective estimates for measures of sets of real numbers with a given order of approximation by quadratic irrationalities. Vestsi Akad. Navuk Belarusi Ser. Fiz.-Mat. Navuk. 1996 (4) (1996), 10–15 (In Russian).Google Scholar
[6] Beresnevich, V. On approximation of real numbers by real algebraic numbers. Acta Arith. 90 (2) (1999), 97–112.Google Scholar
[7] Beresnevich, V. A Groshev type theorem for convergence on manifolds. Acta Math. Hungar. 94 (1–2) (2002), 99–130.Google Scholar
[8] Beresnevich, V. Rational points near manifolds and metric Diophantine approximation. Ann. of Math. (2) 175 (1) (2012), 187–235.Google Scholar
[9] Beresnevich, V., Bernik, V. and Dodson, M. On the Hausdorff dimension of sets of well-approximable points on nondegenerate curves. Dokl. Nats. Akad. Nauk Belarusi. 46 (2002), 18–20.Google Scholar
[10] Beresnevich, V., Bernik, V., Dodson, M. and Velani, S. Classical Metric Diophantine Approximation Revisited. Roth Festschrift - essays in honour of Klaus Roth on the occasion of his 80th birthday. Editors: Chen, W., Gowers, T., Halberstam, H., Schmidt, W.M. and Vaughan, R.C. (Cambridge University Press, 2009), 38–61.Google Scholar
[11] Beresnevich, V. V., Bernik, V. I., Kleinbock, D. Y. and Margulis, G. A. Metric Diophantine approximation: the Khintchine–Groshev theorem for nondegenerate manifolds. Mosc. Math. J. 2 (2) (2002), 203–225. Dedicated to Yuri I. Manin on the occasion of his 65th birthday.Google Scholar
[12] Beresnevich, V., Dickinson, D. and Velani, S. Sets of exact ‘logarithmic’ order in the theory of Diophantine approximation. Math. Ann. 321 (2001), 253–273.Google Scholar
[13] Beresnevich, V., Dickinson, D. and Velani, S. Measure theoretic laws for lim sup sets. Mem. Amer. Math. Soc. 179 (846) (2006), x+91.Google Scholar
[14] 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. With an Appendix II by R. C. Vaughan.Google Scholar
[15] Beresnevich, V. V., Vaughan, R. C. and Velani, S. L. Inhomogeneous Diophantine approximation on planar curves. Math. Ann. 349 (4) (2011), 929–942.Google Scholar
[16] Beresnevich, V. and Velani, S. A mass transference principle and the Duffin–Schaeffer conjecture for Hausdorff measures. Ann. of Math. (2) 164 (3) (2006), 971–992.Google Scholar
[17] Beresnevich, V. and Velani, S. Classical metric diophantine approximation revisited: the Khintchine–Groshev Theorem. International Mathematics Research Notices. 2010 (1) (2010), 69–86.Google Scholar
[18] Beresnevich, V. and Zorin, E. Explicit bounds for rational points near planar curves and metric Diophantine approximation. Adv. Math. 225 (2010), 3064–3087.Google Scholar
[19] Bernik, V. I. Application of the Hausdorff dimension in the theory of Diophantine approximations. Acta Arith. 42 (3) (1983), 219–253.Google Scholar
[20] Bernik, V. I. and Dodson, M. M. Metric Diophantine approximation on manifolds. Cambridge Tracts in Mathematics vol. 137 (Cambridge University Press, Cambridge, 1999).Google Scholar
[21] Bernik, V., Kleinbock, D. and Margulis, G. A. Khintchine-type theorems on manifolds: the convergence case for standard and multiplicative versions. Internat. Math. Res. Notices (9) (2001), 453–486.Google Scholar
[22] Budarina, N., Dickinson, D. and Levesley, J. Simultaneous Diophantine approximation on polynomial curves. Mathematika. 56 (1) (2010), 77–85.Google Scholar
[23] Dickinson, D. and Dodson, M. Extremal manifolds and Hausdorff dimension. Duke Math. J. 101 (2000), 271–281.Google Scholar
[24] Dickinson, D. and Velani, S. L. Hausdorff measure and linear forms. J. Reine Angew. Math. 490 (1997), 1–36.Google Scholar
[25] Groshev, A. V. Une théorème sur les systemes des formes lineaires. Doklady Akad. Nauk SSSR. 19 (1938), 151–152.Google Scholar
[26] Huang, J.-J. Hausdorff theory of dual approximation on planar curves. Pre-print: arXiv:1403.8038.Google Scholar
[27] Hussain, M. and Yusupova, T. On weighted inhomogeneous Diophantine approximation on planar curves. Math. Proc. Camb. Phil. Soc. 154 (2013), 225–241.Google Scholar
[28] Jarník, V. Über die simultanen diophantischen approximationen. Math. Z. 33 (1931), 505–543.Google Scholar
[29] Khintchine, A. Einige Sätze über Kettenbrüche, mit Anwendungen auf die Theorie der Diophantischen Approximationen. Math. Ann. 92 (1–2) (1924), 115–125.Google Scholar
[30] Kleinbock, D. Y. and Margulis, G. A. Flows on homogeneous spaces and Diophantine approximation on manifolds. Ann. of Math. (2) 148 (1) (1998), 339–360.Google Scholar
[31] Mahler, K. Über das Maß der Menge aller S-Zahlen. Math. Ann. 106 (1) (1932), 131–139.Google Scholar
[32] Schmidt, W. M. A metrical theorem in diophantine approximation. Canad. J. Math. 12 (1960), 619–31.Google Scholar
[33] Schmidt, W. M. Metrische Sätze über simultane Approximation abhängiger Grössen. Monatsh. Math. 68 (1964), 154–166.Google Scholar
[34] Sprindžuk, V. G. Mahler's problem in metric number theory (translated from the Russian by Volkmann, B.). Transl. Math. Monogr. vol. 25 (American Mathematical Society, Providence, R.I., 1969).Google Scholar
[35] Vaughan, R. C. and Velani, S. Diophantine approximation on planar curves: the convergence theory. Invent. Math. 166 (1) (2006), 103–124.Google Scholar