In Analysis 3 we learn that the middle third Cantor set has zero Lebesgue measure. The 1-dimensional Lebesgue measure arises by considering the sums of lengths of intervals covering a set. If we instead were to sum the lengths of intervals raised to the power s we would get the s-dimensional Hausdorff measure. When s=\log 2/\log3, the s-dimensional Hausdorff measure of the middle third Cantor set is a positive real number.
We have made a number of implicit assertions: that the Hausdorff measure is a measure is quite tricky, and actually verifying the positivity of the measure of the middle third Cantor set is done in a non-direct way!
In this project we'll study the Hausdorff measure and some fundamental techniques. As an example of what this means: observe that any covering of a 1-dimensional subset by intervals is "easily" turned into a disjoint covering by intervals; whereas for a covering of a subset in 2-d by open balls/ discs we can't just take differences of discs, we have to do something different (namely, use Vitali's lemma).
We will explore further topics where the concept of Hausdorff measure helps us understand "fine structure of subsets" such as Marstrand's slicing theorem and Marstrand's projection theorem.
Pre-requisites: Analysis 3.
Co-requisite: Ergodic Theory 4.
Some references (available online through Durham library):
Fractals in Probability and Analysis, C.J. Bishop and Y. Peres.
Fractal Geometry: Mathematical Foundations and Applications, K.J. Falconer.