Hostname: page-component-8448b6f56d-c4f8m Total loading time: 0 Render date: 2024-04-24T14:51:43.066Z Has data issue: false hasContentIssue false
  • English
  • Français

The Hausdorff Dimension of a Set of Normal Numbers II

Part of: Probabilistic theory: distribution modulo $1$; metric theory of algorithms

Published online by Cambridge University Press:  09 April 2009

A. D. Pollington
A. D. Pollington
Affiliation:
Department of MathematicsBrigham Young University Provo, Utah 84602, U.S.A.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let R, S be a partition of 2, 3,… so that rational powers fall in the same class. Let (λn) be any real sequence; we show that there exists a set N, of dimension 1, so that (x + λn) (n = 1,2, …) are normal to every base from R and to no base from S, for every x ∈ N.

Keywords

normal numbersHausdorff dimensionsum sets

MSC classification

Secondary: 11K16: Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc. 11K55: Metric theory of other algorithms and expansions; measure and Hausdorff dimension
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 44 , Issue 2 , April 1988 , pp. 259 - 264
Copyright
Copyright © Australian Mathematical Society 1988

References

[1]Brown, G., Moran, W., and Pearce, C. E. M., ‘Riesz products and normal numbers’, J. London Math. Soc. (2) 32 (1985), 12–18.CrossRefGoogle Scholar
[2]Pearce, C. E. M. and Keane, M. S., ‘On normal numbers’, J. Austral. Math. Soc. (1) 32 (1982), 79–87.CrossRefGoogle Scholar
[3]Pollington, A. D., ‘The Hausdorff dimension of a set of normal numbers’, Pacific J. Math. 95 (1980) 193–204.CrossRefGoogle Scholar
[4]Schmidt, W. M., ‘Über die Normalität von Zahlen zu verschiedenen Basen’, Acta Arith. 7 (1961–1962), 299–309.CrossRefGoogle Scholar