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The Hausdorff dimension of systems of simultaneously small linear forms

Part of: Diophantine approximation, transcendental number theory

Published online by Cambridge University Press:  26 February 2010

H. Dickinson
H. Dickinson
Affiliation:
Department of Mathematics, The University of YorkHeslington, York, YO1 5DD
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Extract

In this paper the Hausdorff dimension of systems of real linear forms which are simultaneously small for infinitely many integer vectors is determined. A system of real linear forms,

where ai, xij∈ℝ, 1 ≤i≤m, 1≤j≤n will be denoted more concisely as

where a∈⇝m, X∈ℝmn and ℝmn is identified with Mm × n(ℝ), the set of real m × n matrices. The supremum norm of any vector in k dimensional Euclidean space, ℝk will be denoted by |v|. The distance of a point a from a set B, will be denoted by dist (a, B) = inf {|a − b|: b ∈ B}.

MSC classification

Secondary: 11J13: Simultaneous homogeneous approximation, linear forms
Type
Research Article
Information
Mathematika , Volume 40 , Issue 2 , December 1993 , pp. 367 - 374
Copyright
Copyright © University College London 1993

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