Assouad dimensions and local structure of fractal sets and measures

The roots of geometry are in developing mathematical tools necessary for land surveying, construction, astronomy and various crafts. In these fields, it is often crucial to precisely measure distances, lengths, areas or volumes of various shapes and figures. However, many shapes, figures or processes found in nature, different fields of engineering or physics are too complicated to be precisely measured with tools from classical geometry. Especially problematic are fractals – objects which contain intricate detail at all scales – such as ice crystals, stock market fluctuations or mountain ranges.

It is often difficult or impossible to describe the size of a fractal precisely with tools from classical geometry. For example, often the measurements for the length or area of a fractal grow or decrease, respectively, as the precision of the measurement increases. Fractal geometry is a field of mathematics specialised in developing tools for describing the size, complexity and other geometric features of fractals. The philosophical objective in fractal geometry is to develop these tools, understand them by studying the structure of prototypical fractal models and apply them to compare and classify naturally arising fractals from applied fields or other fields of mathematics. For example, according to some medical studies, tools from fractal geometry can be used to identify certain cancer cells, which have a rougher structure than healthy cells, with more precision than classical methods.

Often, fractals have a self-similar structure: they consist of, possibly distorted or overlapping, copies of themselves at all small scales. In my doctoral dissertation, I study the local structure of fractals, which is easily affected by these overlaps and distortions. I introduce a novel method for quantifying the local size of fractal mass distributions and compare this method to previous methods. I show that for many self-similar fractals, this method uncovers a fine structure, which earlier methods are unable to identify. In addition, I study self-similar fractals, which contain both distortions and overlaps, and show that the distortion leads to a comb-like local structure in the fractal at small scales. This structure can then be used, for example, to distinguish fractals which look similar at large scales.