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ON SHRINKING TARGETS FOR ℤm ACTIONS ON TORI

Part of: Diophantine approximation, transcendental number theory Low-dimensional dynamical systems Probabilistic theory: distribution modulo $1$; metric theory of algorithms

Published online by Cambridge University Press:  16 April 2010

Yann Bugeaud ,
Stephen Harrap ,
Simon Kristensen  and
Sanju Velani
Yann Bugeaud
Affiliation:
Département de Mathématiques, Université de Strasbourg, 7, rue René Descartes, F-67084 Strasbourg cedex, France (email: bugeaud@math.unistra.fr)
Stephen Harrap
Affiliation:
Department of Mathematics, University of York, Heslington, York YO10 5DD, U.K. (email: sgh111@york.ac.uk)
Simon Kristensen
Affiliation:
Department of Mathematical Sciences, Faculty of Science, Aarhus University, Ny Munkegade 118, DK-8000 Aarhus C, Denmark (email: sik@imf.au.dk)
Sanju Velani
Affiliation:
Department of Mathematics, University of York, Heslington, York YO10 5DD, U.K. (email: slv3@york.ac.uk)
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Abstract

Let A be an n×m matrix with real entries. Consider the set BadA of x∈[0,1)n for which there exists a constant c(x)>0 such that for any q∈ℤm the distance between x and the point {Aq} is at least c(x)|q|−m/n. It is shown that the intersection of BadA with any suitably regular fractal set is of maximal Hausdorff dimension. The linear form systems investigated in this paper are natural extensions of irrational rotations of the circle. Even in the latter one-dimensional case, the results obtained are new.

MSC classification

Secondary: 11K60: Diophantine approximation 11J83: Metric theory 37E10: Maps of the circle 37E45: Rotation numbers and vectors
Type
Research Article
Information
Mathematika , Volume 56 , Issue 2 , July 2010 , pp. 193 - 202
Copyright
Copyright © University College London 2010

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