A note on equivalent interval covering systems for Hausdorff dimension on ℝ

A note on equivalent interval covering systems for Hausdorff dimension on ℝ

The Hausdorff dimension of a set in ℝ is usually defined by considering countable coverings of the set by general intervals. In this note we establish sufficient conditions under which coverings whose members are restricted to a particular family g of intervals will produce the same value for dimens...

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Bibliographic Details
Main Author: C. D. Cutler
Format: Article
Language:English
Published: Hindawi Limited 1988-01-01
Series:International Journal of Mathematics and Mathematical Sciences
Subjects:
Hausdorff dimension
Vitali covering
self-similar.
Online Access:http://dx.doi.org/10.1155/S016117128800078X
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spelling doaj-93ad7e2ba1534d8fbbf4171874f250fb2020-11-25T00:32:09ZengHindawi LimitedInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251988-01-0111464364910.1155/S016117128800078XA note on equivalent interval covering systems for Hausdorff dimension on ℝC. D. Cutler0Department of Statistics and Actuarial Science, University of Waterloo, Ontario, Waterloo N2L 3G1, CanadaThe Hausdorff dimension of a set in ℝ is usually defined by considering countable coverings of the set by general intervals. In this note we establish sufficient conditions under which coverings whose members are restricted to a particular family g of intervals will produce the same value for dimension. A result of Billingsley is then employed to obtain a general technique for computing the dimensions of sets defined by certain types of generalized expansions. A specific example is included.http://dx.doi.org/10.1155/S016117128800078XHausdorff dimensionVitali coveringself-similar.
collection DOAJ
language English
format Article
sources DOAJ
author C. D. Cutler
spellingShingle C. D. Cutler
A note on equivalent interval covering systems for Hausdorff dimension on ℝ
International Journal of Mathematics and Mathematical Sciences
Hausdorff dimension
Vitali covering
self-similar.
author_facet C. D. Cutler
author_sort C. D. Cutler
title A note on equivalent interval covering systems for Hausdorff dimension on ℝ
title_short A note on equivalent interval covering systems for Hausdorff dimension on ℝ
title_full A note on equivalent interval covering systems for Hausdorff dimension on ℝ
title_fullStr A note on equivalent interval covering systems for Hausdorff dimension on ℝ
title_full_unstemmed A note on equivalent interval covering systems for Hausdorff dimension on ℝ
title_sort note on equivalent interval covering systems for hausdorff dimension on ℝ
publisher Hindawi Limited
series International Journal of Mathematics and Mathematical Sciences
issn 0161-1712
1687-0425
publishDate 1988-01-01
description The Hausdorff dimension of a set in ℝ is usually defined by considering countable coverings of the set by general intervals. In this note we establish sufficient conditions under which coverings whose members are restricted to a particular family g of intervals will produce the same value for dimension. A result of Billingsley is then employed to obtain a general technique for computing the dimensions of sets defined by certain types of generalized expansions. A specific example is included.
topic Hausdorff dimension
Vitali covering
self-similar.
url http://dx.doi.org/10.1155/S016117128800078X
work_keys_str_mv AT cdcutler anoteonequivalentintervalcoveringsystemsforhausdorffdimensiononr
AT cdcutler noteonequivalentintervalcoveringsystemsforhausdorffdimensiononr
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