The Hausdorff dimension of some dynamically defined sets

Abstract

The topic of this thesis is the analysis of the dimension of various fractal sets and their associated measures. This thesis is structured into five chapters. The last three each establish
new results on three core topics that are thematically linked, and the first two chapters
serve as an introduction and technical background for these linked topics.
Chapter 1 introduces the framework of the thesis: the interplay between size (quantified
through notions such as Hausdorff dimension) and dynamics in mathematical systems. By studying the simplest examples of the objects considered in this thesis, we aim to illustrate the nature
of the problems under investigation. Chapter 2 contextualizes these topics by introducing the historical results, developing the relevant theoretical foundations, and outlining general pathways
to the central questions explored in later chapters.
The final three chapters each treat a separate problem within the general area introduced in
Chapters 1–2. Chapter 3 investigates the Hausdorff dimension of eventually always hitting sets
in self-conformal settings. These sets are a counterpart to the notion of infinitely recurring sets
and serve as liminf counterparts to shrinking target sets. In Chapter 4, the focus shifts to random
Bernoulli convolution measures, where two key properties are analyzed: the existence of interior
points within their support and the polynomial decay rate of their Fourier transforms. Finally,
Chapter 5 is devoted to proving that the set of ω-badly approximable points and the set of ω-well
approximable points share the same Hausdorff dimension in Rd.

Date of Award	30 Sept 2025
Original language	English
Awarding Institution	University of Bristol
Supervisor	Nina C Snaith (Supervisor) & Henna L L Koivusalo (Supervisor)