Jonathan Fraser - Research

I work in analysis but with most of my research concerning fractal geometry. I am also interested in connections between fractal geometry and wider mathematics, especially geometric measure theory, Fourier analysis, dynamical systems, and probability theory. Roughly speaking, a fractal is an object or process which exhibits complexity over a large range of scales, see some examples below. An important way to characterise fractals is via their dimension. In fact, it is already a fascinating problem to define the dimension of a fractal.

Here are some of the more specific research topics I am interested in. Follow the links to find out more!

Self-affine sets and measures - self-affine carpets, multifractals

Projections of fractals - Marstrand type theorems and applications

The distance set problem - a notorious open problem in GMT

Conformal dynamics - Kleinian groups, Julia sets, conformal IFSs

Fourier analysis - Fourier analytic techniques in GMT, Fourier dimension

Dimension interpolation - Assouad spectrum, intermediate dimensions, Fourier spectrum

Arithmetic combinatorics - sumsets, (approximation of) arithmetic progressions in fractals and sets of integers

Inhomogeneous IFSs - beautiful variant of standard IFS due to Barnsley and Demko

Self-similar sets and measures - self-similar sets with overlaps

Left: a self-affine carpet of the type introduced by Bedford and McMullen. Its lower, Hausdorff, box, and Assouad dimension are all distinct. Centre: the Apollonian circle packing. It can be realised as the limit set of a Kleinian group. Right: the graph of the popcorn function. Together with Haipeng Chen and Han Yu I proved that it has box dimension 4/3.