Open Collections

Cruz, Angel Diosdado

University of British Columbia

This thesis addresses two distinct problems in geometric measure theory, both involving point configurations. The first is concerned with sets of positive Lebesgue measure that avoid affine copies of infinite configurations. The second is concerned with the Fourier dimensions of sets and the configurations they contain. When considering positive Lebesgue measure, we construct a fractal set and show how this construction avoids fast-decaying sequences. Before proving this result, we present some other constructions and strategies to address avoidance in sets of positive Lebesgue measure. Regarding the Fourier dimension, we not only consider avoidance but provide quantitative results that directly connect the Fourier dimension of a set to the inclusion of certain algebraic configurations within the set. We first introduce a technique that was introduced by Yiyu Liang and Malabika Pramanik, which is then generalized and applied to achieve the main results.

Text

2025-06-23

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/91412

Mathematics

University of British Columbia

UBCV

http://creativecommons.org/licenses/by-nc-nd/4.0/