dc.contributor.author Ng, Ka Shing dc.date.accessioned 2014-03-28 19:34:19 (GMT) dc.date.available 2014-03-28 19:34:19 (GMT) dc.date.issued 2014-03-28 dc.date.submitted 2014 dc.identifier.uri http://hdl.handle.net/10012/8300 dc.description.abstract For every positive, decreasing, summable sequence $a=(a_i)$, we can construct a Cantor set $C_a$ associated with $a$. These Cantor sets are not necessarily self-similar. Their dimensional properties and measures have been studied in terms of the sequence $a$. en In this thesis, we extend these results to a more general collection of Cantor sets. We study their Hausdorff and packing measures, and compare the size of Cantor sets with the more refined notion of dimension partitions. The properties of these Cantor sets in relation to the collection of cut-out sets are then considered. The multifractal spectrum of $\mathbf{p}$-Cantor measures on these Cantor sets are also computed. We then focus on the special case of homogeneous Cantor sets and obtain a more accurate estimate of their exact measures. Finally, we prove the $L^p$-improving property of the $\mathbf{p}$-Cantor measure on a homogeneous Cantor set as a convolution operator. dc.language.iso en en dc.publisher University of Waterloo en dc.subject Cantor sets en dc.subject cut-out sets en dc.subject Hausdorff measures en dc.subject packing measures en dc.subject dimension partition en dc.subject multifractal analysis en dc.subject L^p improving en dc.subject gauge functions en dc.title Some aspects of Cantor sets en dc.type Doctoral Thesis en dc.pending false dc.subject.program Pure Mathematics en uws-etd.degree.department Pure Mathematics en uws-etd.degree Doctor of Philosophy en uws.typeOfResource Text en uws.peerReviewStatus Unreviewed en uws.scholarLevel Graduate en
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