PhD thesis defence seminar

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.