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Diophantine approximation and a lower bound for Hausdorff dimension

Published online by Cambridge University Press:  26 February 2010

M. M. Dodson ,
B. P. Rynne  and
J. A. G. Vickers
M. M. Dodson
Affiliation:
Department of Mathematics, University of York, York, YO1 5DD
B. P. Rynne
Affiliation:
Department of Mathematics, University of Dundee, Dundee, DD1 4HN
J. A. G. Vickers
Affiliation:
Department of Mathematics, University of Southampton, Southampton, SO9 5NH
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Extract

Sets of the general form

where U is a subset of ℛk and is a family of subsets of U indexed by a set J, are common in the theory of Diophantine approximation [4, 7, 18, 19]. They are also closely connected with exceptional sets arising in analysis and with sets of “small divisors” in dynamical systems [1, 8, 15”. When J is the set of positive integers ℕ, the set Λ(ℱ) is of course the lim-sup of the sequence of sets Fj, j = 1, 2,… [11, p. 1]. We will also call sets of the form (1), with the more general index set J, lim-sup sets. When such lim-sup sets have Lebesgue measure zero, it is of interest to determine their Hausdorff dimension. It is usually difficult to obtain a good lower bound for the Hausdorff dimension (and it can be much harder to determine than an upper bound). In this paper we will obtain a lower bound for the dimension of lim-sup sets of the form (1) for a fairly general class of families ℕ which includes a range of results in the theory of Diophantine approximation. This lower bound depends explicitly on the geometric structure and distribution in U of the sets Fα in ℕ.

Type
Research Article
Information
Mathematika , Volume 37 , Issue 1 , June 1990 , pp. 59 - 73
Copyright
Copyright © University College London 1990

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