Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-26T08:37:08.492Z Has data issue: false hasContentIssue false
  • English
  • Français

On Diophantine transference principles

Published online by Cambridge University Press:  20 February 2018

ANISH GHOSH  and
ANTOINE MARNAT
ANISH GHOSH
Affiliation:
School of Mathematics, Tata Institute of Fundamental Research, Mumbai, India400005. e-mail: [email protected]
ANTOINE MARNAT
Affiliation:
Institute of Analysis and Number theory, Technische Universität Graz, 8010 Garz, Austria. Department of Mathematics, University of York, Heslington, York, YO10 5DD, UK. e-mails: [email protected], [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We provide an extension of the transference results of Beresnevich and Velani connecting homogeneous and inhomogeneous Diophantine approximation on manifolds and provide bounds for inhomogeneous Diophantine exponents of affine subspaces and their nondegenerate submanifolds.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 166 , Issue 3 , May 2019 , pp. 415 - 431
Copyright
Copyright © Cambridge Philosophical Society 2018 

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.)

Footnotes

†

Supported by an ISF-UGC grant.

‡

Supported by the Austrian Science Fund (FWF), Project F5510-N26 and START-project Y-901.

References

REFERENCES

[1] Aka, M., Breuillard, E., Rosenzweig, L., and de Saxće, N. On metric Diophantine approximation in matrices and Lie groups. C. R. Math. Acad. Sci. Paris 353 (2015), no. 3, 185–189.10.1016/j.crma.2014.12.007Google Scholar
[2] Beresnevich, V., Kleinbock, D. and Margulis, G. Non-planarity and metric Diophantine approximation for systems of linear forms. J. Théor. Nombres Bordeaux 27 (2015), no. 1, 1–31.10.5802/jtnb.890Google Scholar
[3] Beresnevich, V. and Velani, S. An inhomogeneous transference principle and Diophantine approximation. Proc. Lond. Math. Soc. (3) 101 (2010), no. 3, pp. 821–851.10.1112/plms/pdq002Google Scholar
[4] Beresnevich, V. and Velani, S. Simultaneous inhomogeneous Diophantine approximations on manifolds. Fundam. Prikl. Mat. 16 (2010), no. 5, pp. 3–17.Google Scholar
[5] Bugeaud, Y. and Laurent, M. On exponents of homogeneous and inhomogeneous Diophantine approximation. Mosc. Math. J. 5 (2005), no. 4, pp. 747–766, 972.10.17323/1609-4514-2005-5-4-747-766Google Scholar
[6] Das, T., Fishman, L., Simmons, D. and Urbanski, M. Extremality and dynamically defined measures, I: Diophantine properties of quasi-decaying measures. http://arxiv.org/abs/1504.04778, to appear in Selecta Mathematica.Google Scholar
[7] German, O. On Diophantine exponents and Khintchine's transference principle. Mosc. J. Comb. Number Theory 2 (2012), no. 2, 22–51.Google Scholar
[8] German, O. Transference inequalities for multiplicative Diophantine exponents. Tr. Mat. Inst. Steklova 275 (2011), pp. 227–239.Google Scholar
[9] Ghosh, A. Diophantine approximation on subspaces of ℝn and dynamics on homogeneous spaces. To appear in Handbook of Group Actions Vol III/IV, Editors Lizhen Ji, Athanase Papadopoulos, Shing–Tung Yau.Google Scholar
[10] Kleinbock, D. Extremal subspaces and their submanifolds. Geom. Funct. Anal 13 (2003), No 2, pp. 437–466.10.1007/s000390300011Google Scholar
[11] Kleinbock, D. An extension of quantitative nondivergence and applications to Diophantine exponents. Trans. Amer. Math. Soc. 360 (2008), no. 12, pp. 6497–6523.10.1090/S0002-9947-08-04592-3Google Scholar
[12] Kleinbock, D. and Margulis, G. A. Flows on homogeneous spaces and Diophantine Approximation on Manifolds. Ann Math 148 (1998), pp. 339–360.10.2307/120997Google Scholar
[13] Kleinbock, D. Y., Margulis, G. A. and Wang, J. Metric Diophantine approximation for systems of linear forms via dynamics. Int. J. Number Theory 6 (2010), no. 5, 1139–168.10.1142/S1793042110003423Google Scholar
[14] Kleinbock, D., Lindenstrauss, E. and Weiss, B. On fractal measures and Diophantine approximation. Selecta Math. (N.S.), 10 (2004), pp. 479–523.10.1007/s00029-004-0378-2Google Scholar
[15] Zhang, Y. Diophantine exponents of affine subspaces: the simultaneous approximation case. J. Number Theory 109 (2009) pp. 1976–1989.10.1016/j.jnt.2009.02.013Google Scholar
[16] Zhang, Y. Multiplicative Diophantine exponents of hyperplanes and their nondegenerate submanifolds. J. Reine Angewandte Mathe. 2012 (2012), issue 664, pp. 93–113.Google Scholar