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# PhD thesis defence seminar

Thursday, March 6, 2014 — 2:30 PM EST

## Michael Ka Shing Ng, Pure Mathematics, University of Waterloo

### "Some aspects of Cantor sets"

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 $a$.

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. At the end, we prove the $L^p$-improving property of
the $\mathbf{p}$-Cantor measure on a homogeneous Cantor set as a
convolution operator.

Location
MC - Mathematics & Computer Building
5136B
200 University Avenue West

Waterloo, ON N2L 3G1

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