2026-09-22T14:35:44Z https://www.ideals.illinois.edu/oai-pmh

oai:www.ideals.illinois.edu:2142/121393 2023-12-13 col_2142_5131 col_2142_16340 com_2142_5130 com_2142_16339 com_2142_8903

Hinkkanen, Aimo Tyson, Jeremy Erdogan, Burak Hildebrand, AJ 2023-08 application/pdf en text Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2023-12-04 without embargo terms The student, Efstathios Konstantinos Chrontsios Garitsis, accepted the attached license on 2023-04-30 at 21:10. The student, Efstathios Konstantinos Chrontsios Garitsis, submitted this Dissertation for approval on 2023-04-30 at 21:16. This Dissertation was approved for publication on 2023-05-04 at 17:09. DSpace SAF Submission Ingestion Package generated from Vireo submission #19268 on 2023-12-04 at 16:59:37 Quasiconformal mappings and fractal geometry Chrontsios Garitsis, Efstathios Konstantinos 2023-05-04 Quasiconformal Fractal Geometry Analysis 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 $|\sum_{n\leq x}f(n)e^{2\pi i n t}|$ grows at a specific uniform rate, then the Fourier series $\sum_{n=1}^\infty \frac{f(n)}{n}e^{2\pi i n t}$ 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 $K$. Text eng https://hdl.handle.net/2142/121393 Copyright 2023 Efstathios Konstantinos Chrontsios Garitsis Ph.D. Dissertation Mathematics University of Illinois at Urbana-Champaign Mathematics