The spectral theory of regular sequences

The regular sequences are a natural generalisation of automatic sequences - sequences which are the fixed points of uniform substitution systems. A number of measure theoretic methods can be used to study automatic sequences, however these are not available for regular sequences, motivating the search for a useful method of associating measures to them. Inspired particularly by the diffraction measure of automatic sequences and their fractal properties, Baake and Coons recently developed the concept of the ghost measure of a regular sequence. This thesis develops the basic theory of these ghost measures. We begin with a detailed study of the example class of affine 2-regular sequences. This leads to a number of observations and questions about the Lebesgue decomposition of ghost measured. Motivated by these questions we develop further the techniques that proved useful into general tools for the study of ghost measures and use them to prove theorems about the existence of the ghost measure under certain assumptions. Building on these general tools, a major theoretical result about the purity of the Lebesgue decomposition of certain ghost measures is proven. Finally, we apply all of these tools to study the connections between fractals and the ghost measures of regular sequences.