Description

For example, the famous Cantor middle-third set is obtained by starting from the interval \([0,1]\), then removing the middle-third to obtain two intervals, then removing the middle-third of each of these and so on. The Cantor middle-third set is a subset of the interval \([0,1]\) and has 1-dimensional Lebesgue measure zero.

On the other hand, there is a dimension for the Cantor middle-third set in which the corresponding Hausdorff measure is finite and non-zero! This dimension is called the Hausdorff dimension. For the Cantor middle-third set the Hausdorff dimension is \( \log 2/ \log 3\).

In this project we will first study the definitions and some properties of Hausdorff measure and Hausdorff dimension. Following this, some fascinating directions that the project could develop into include (but are not limited to):

• Fractals and self-similar curves. In general, it is difficult to compute or estimate the Hausdorff dimension and Hausdorff measure of a set. The fact that the Cantor middle-third set has some sort of self-replicating behaviour makes it possible to compute the Hausdorff dimension explicitly. Similarly, the Hausdorff dimension of other sets with self-similar behaviour can be computed explicitly, e.g. that of the von Koch snowflake.
• Space-filling curves. A construction due to Peano gives a continuous curve that fills the square \([0,1] \times [0,1] \), despite the fact that an interval is 1-dimensional and the square is 2-dimensional. This intriguing construction gives a map from the interval \([0,1] \) to the square \([0,1] \times [0,1] \) that preserves the Lebesgue measure.
• Regularity and Besicovitch sets. The Euclidean space of dimension \(d \geq 3\) enjoys a certain regularity property that does not hold in dimension 2. In the planar case, there exist special sets called Besicovitch (or Kakeya) sets that obstruct the regularity property.

Prior and Companion modules

Essential prior modules: Complex Analysis II.

Essential companion modules: Analysis III.

Depending on your preferred direction of the project, some of the following modules could be useful companions but are not essential: Differential Geometry III, Geometry III.

Some Resources

K. J. Falconer Fractal geometry : mathematical foundations and applications. Chichester : Wiley, c1993.

G. B. Folland Real analysis : modern techniques and their applications. Second edition. Pure and applied mathematics. New York : John Wiley & Sons, Inc., 1999.

C. A. Rogers Hausdorff measures. Cambridge University Press, 1970.

E. M. Stein and R. Shakarchi Real Analysis : Measure Theory, Integration, and Hilbert Spaces. Princeton, NJ : Princeton University Press, 2009.