Hostname: page-component-848d4c4894-ndmmz Total loading time: 0 Render date: 2024-05-22T01:34:33.062Z Has data issue: false hasContentIssue false
  • English
  • Français

A Hausdorff measure version of the Jarník–Schmidt theorem in Diophantine approximation

Published online by Cambridge University Press:  05 April 2017

DAVID SIMMONS
DAVID SIMMONS*
Affiliation:
University of York, Department of Mathematics, Heslington, York YO10 5DD. e-mail: David.Simmons@york.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

We solve the problem of giving sharp asymptotic bounds on the Hausdorff dimensions of certain sets of badly approximable matrices, thus improving results of Broderick and Kleinbock (preprint 2013) as well as Weil (preprint 2013), and generalising to higher dimensions those of Kurzweil ('51) and Hensley ('92). In addition we use our technique to compute the Hausdorff f-measure of the set of matrices which are not ψ-approximable, given a dimension function f and a function ψ : (0, ∞) → (0, ∞). This complements earlier work by Dickinson and Velani ('97) who found the Hausdorff f-measure of the set of matrices which are ψ-approximable.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 164 , Issue 3 , May 2018 , pp. 413 - 459
Copyright
Copyright © Cambridge Philosophical Society 2017 

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] Beresnevich, V. and Velani, S. Classical metric Diophantine approximation revisited: the Khintchine–Groshev theorem. Int. Math. Res. Not. IMRN (2010), no. 1, 69–86.Google Scholar
[2] Bernik, V. and Dodson, M. Metric Diophantine approximation on manifolds. Cambridge Tracts in Mathematics, vol. 137 (Cambridge University Press, Cambridge, 1999).CrossRefGoogle Scholar
[3] Besicovitch, A. S. Sets of fractional dimensions (IV): on rational approximation to real numbers. J. London Math. Soc. 9 (1934), no. 2, 126–131.CrossRefGoogle Scholar
[4] Bovey, J. and Dodson, M. The Hausdorff dimension of systems of linear forms. Acta Arith. 45 (1986), no. 4, 337–358.CrossRefGoogle Scholar
[5] Broderick, R., Fishman, L., Kleinbock, D., Reich, A. and Weiss, B. The set of badly approximable vectors is strongly C 1 incompressible. Math. Proc. Camb. Phil. Soc. 153 (2012), no. 02, 319–339.CrossRefGoogle Scholar
[6] Broderick, R., Fishman, L. and Simmons, D. Badly approximable systems of affine forms and incompressibility on fractals. J. Number Theory. 133 (2013), no. 7, 2186–2205.CrossRefGoogle Scholar
[7] Broderick, R. and Kleinbock, D. Dimension estimates for sets of uniformly badly approximable systems of linear forms. Int. J. Number Theory 11 (2015), no. 7, 2037–2054.CrossRefGoogle Scholar
[8] Bugeaud, Y. Approximation by algebraic numbers. Cambridge Tracts in Mathematics. vol. 160 (Cambridge University Press, Cambridge, 2004).CrossRefGoogle Scholar
[9] Cassels, J. W. S. An Introduction to the Geometry of Numbers. Corrected reprint of the 1971 edition Classics in Mathematics (Springer-Verlag, Berlin, 1997).Google Scholar
[10] Dani, S. G. Bounded orbits of flows on homogeneous spaces. Comment. Math. Helv. 61 (1986), no. 4, 636–660.CrossRefGoogle Scholar
[11] Dickinson, D. and Velani, S. Hausdorff measure and linear forms. J. Reine Angew. Math. 490 (1997), 1–36.Google Scholar
[12] Fishman, L. Schmidt's game, badly approximable matrices and fractals. J. Number Theory 129 (2009), no. 9, 2133–2153.CrossRefGoogle Scholar
[13] Fishman, L. Schmidt's game on fractals. Israel J. Math. 171 (2009), no. 1, 77–92.CrossRefGoogle Scholar
[14] Fishman, L., Simmons, D. and Urbański, M. Diophantine approximation in Banach spaces. J. Théor. Nombres Bordeaux 26 (2014), no. 2, 363–384.CrossRefGoogle Scholar
[15] Groshev, A. V. Un théorème sur les systèmes des formes linéaires (a theorem on a system of linear forms). Dokl. Akad. Nauk SSSR 19 (1938), 151–152 (Russian).Google Scholar
[16] Harman, G. Metric number theory. London Math. Soc. Monogs. N.S., 18 (The Clarendon Press, Oxford University Press, New York, 1998).CrossRefGoogle Scholar
[17] Hensley, D. Continued fraction Cantor sets, Hausdorff dimension and functional analysis. J. Number Theory 40 (1992), no. 3, 336–358.CrossRefGoogle Scholar
[18] Howe, R. E. and Moore, C. C. Asymptotic properties of unitary representations. J. Funct. Anal. 32 (1979), no. 1, 72–96.CrossRefGoogle Scholar
[19] Jarník, V. Diophantische Approximationen und Hausdorffsches Mass. Mat. Sb. 36 (1929), 371–382 (German).Google Scholar
[20] Khinchin, A. Y. Einige Sätze über Kettenbrüche, mit Anwendungen auf die Theorie der Diophantischen Approximationen. Math. Ann. 92 (1924), 115–125 (German).Google Scholar
[21] Khinchin, A. Y. Zur metrischen Theorie der diophantischen Approximationen. Math. Zeitschrift 24 (1926), 706–713 (German).CrossRefGoogle Scholar
[22] Kleinbock, D. and Margulis, G. Logarithm laws for flows on homogeneous spaces. Invent. Math. 138 (1999), no. 3, 451–494.CrossRefGoogle Scholar
[23] Kristensen, S., Thorn, R. and Velani, S. Diophantine approximation and badly approximable sets. Advances in Math. 203 (2006), 132–169.CrossRefGoogle Scholar
[24] Kurzweil, J. A contribution to the metric theory of diophantine approximations. Czechoslovak Math. J. 1 (76) (1951), 149–178.CrossRefGoogle Scholar
[25] Mauldin, R. D., Szarek, T. and Urbański, M. Graph directed Markov systems on Hilbert spaces. Math. Proc. Cam. Phil. Soc. 147 (2009), 455–488.CrossRefGoogle Scholar
[26] Moreira, C. Geometric properties of the Markov and Lagrange spectra. http://w3.impa.br/~gugu/Geometric_Properties.pdf, preprint.Google Scholar
[27] Perron, O. Über diophantische Approximationen. Math. Ann. 83 (1921), 77–84 (German).CrossRefGoogle Scholar
[28] Schmidt, W. M. On badly approximable numbers and certain games. Trans. Amer. Math. Soc. 123 (1966), 27–50.CrossRefGoogle Scholar
[29] Schmidt, W. M. Badly approximable systems of linear forms. J. Number Theory 1 (1969), 139–154.CrossRefGoogle Scholar
[30] Weil, S. Jarník-type inequalities. Proc. Lond. Math. Soc. (3) 110 (2015), no. 1, 172–212.CrossRefGoogle Scholar
[31] Winitzki, S. Linear algebra via exterior products. lulu.com (2010).Google Scholar