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Regular and ubiquitous systems, and ℳs∞-dense sequences

Part of: Classical measure theory

Published online by Cambridge University Press:  26 February 2010

Bryan P. Rynne
Bryan P. Rynne
Affiliation:
Department of Mathematics, Heriot-Watt University, Riccarton, Edinburgh. EH14 4AS.
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Abstract

“Regular systems” of numbers in ℝ and “ubiquitous systems” in ℝk, k ≥ 1, have been used previously to obtain lower bounds for the Hausdorff dimension of various sets in ℝ and ℝk respectively. Both these concepts make sense for systems of numbers in ℝ, but the definitions of the two types of object are rather different. In this paper it will be shown that, after certain modifications to the definitions, the two concepts are essentially equivalent.

We also consider the concept of a ℳs∞-dense sequence in ℝk, which was introduced by Falconer to construct classes of sets having “large intersection”. We will show that ubiquitous systems can be used to construct examples of ℳs∞-dense sequences. This provides a relatively easy means of constructing ℳs∞-dense sequences; a direct construction and proof that a sequence is ℳs∞-dense is usually rather difficult.

MSC classification

Secondary: 28A78: Hausdorff and packing measures
Type
Research Article
Information
Mathematika , Volume 39 , Issue 2 , December 1992 , pp. 234 - 243
Copyright
Copyright © University College London 1992

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References

1.Baker, A. and Schmidt, W. M.. Diophantine approximation and Hausdorff dimension. Proc. Lond. Math. Soc., 21 (1970), 1–11.CrossRefGoogle Scholar
2.Baker, R. C.. Dirichlet's theorem on Diophantine approximation. Math. Proc. Camb. Phil. Soc., 83 (1978), 37–59.CrossRefGoogle Scholar
3.Bernik, V. I.. Applications of measure theory and Hausdorff dimension to the theory of Diophantine approximation. In New Advances in Transcendence Theory, ed. Baker, A. (Cambridge University Press, 1988), 25–36.CrossRefGoogle Scholar
4.Dodson, M. M.Rynne, B. P. and Vickers, J. A. G.. Metric Diophantine approximation and Hausdorff dimension on manifolds. Math. Proc. Camb. Phil. Soc., 105 (1989), 547–558.CrossRefGoogle Scholar
5.Dodson, M. M.Rynne, B. P. and Vickers, J. A. G.. Diophantine approximation and a lower bound for Hausdorff dimension. Mathematika, 37 (1990), 59–73.CrossRefGoogle Scholar
6.Falconer, K. J.. Classes of sets with large intersection. Mathematika, 32 (1985), 191–205.CrossRefGoogle Scholar
7.Falconer, K. J.. Fractal Geometry (Wiley, Chichester, 1990).Google Scholar