Dimensional properties of random covering sets

Fractals are objects that exhibit interesting details at arbitrarily small scales. Fractals are often too complicated to be studied by classical methods, and to study fractals, the field of mathematics known as fractal geometry has been developed. One central problem in the field of fractal geometry is to study the size of fractals. A frequently occurring phenomenon for fractals is that familiar notions describing size, such as length, area and volume, do not provide sufficient information about the size of fractals. For example, in the plane there exist many fractals with infinite length and zero area. All these fractals are thus indistinguishable from each other when considering only their length and area, even though some of them clearly appear to be larger than some others. In these kinds of scenarios, the notion of fractal dimension is often useful, as it can be used to provide more refined information on the size of these fractals.

In my doctoral thesis, I study the dimension of a class of random fractals known as random covering sets. Roughly speaking, random covering sets can be formed by first choosing some sequence of shapes (for example, balls or rectangles in the plane) and then moving these shapes around randomly. The random covering set then consists of those points (in the plane), which are covered by infinitely many of these randomly placed shapes. The research problem is then to study how large these random covering sets typically are in terms of dimension. The setting of this question can be varied by changing the ambient space, by choosing different shapes, or by changing the meaning of the word “randomly”, and the nature of the problem depends heavily on the setting. One particularly interesting setting, which is the main focus of the thesis, happens when the probability measure responsible for the randomness possesses some fractal properties.