Quasiconformal mappings and fractal geometry

This thesis discusses three different projects regarding fractals and quasiconformal mappings. The first project involves certain fractal Fourier series motivated by exponential sums studied in areas of number theory. We showed with Hildebrand that if for a function $f:\N\rightarrow \C$ the sum |\sum_{n\leq x}f(n)e^{2π i n t}| grows at a specific uniform rate, then the Fourier series \sum_n=1^\infty \frac{f(n)}{n}e^{2π i n t} is H\"older continuous, which provides upper bounds on the box-counting dimension of fractal sets associated with the series. The second project investigates the quasiconformal distortion of certain dimension notions and spectra. More specifically, we showed with Tyson that a quasiconformal map between domains in Euclidean spaces cannot change the dimensions of sets uncontrollably. This was previously known for the Hausdorff and box-counting dimensions, but our result applies on the Assouad dimension and spectrum, with the latter being the first result of its kind. The third project is motivated by the geometric definition of quasiconformality on the complex plane and provides interesting properties for certain quadrilaterals. Namely, we showed with Hinkkanen that every quadrilateral $Q$ of modulus in some interval $[1/K,K]$, $K>1$, needs to contain a disk of radius comparable to the maximum of the internal distances of $Q$, where the comparability constant only depends on $K$.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2023

Author dc:creator

• Chrontsios Garitsis, Efstathios Konstantinos

Contributors dc:contributor

• Hinkkanen, Aimo
• Tyson, Jeremy
• Erdogan, Burak
• Hildebrand, AJ

• Quasiconformal
• Fractal
• Geometry
• Analysis

Statement dc:rights

• Copyright 2023 Efstathios Konstantinos Chrontsios Garitsis

Language dc:language

en, eng

Handle dc:identifier

https://hdl.handle.net/2142/121393

OAI record oai:identifier

https://www.ideals.illinois.edu/oai-pmh?verb=GetRecord&metadataPrefix=etdms&identifier=oai%3Awww.ideals.illinois.edu%3A2142%2F121393