2026-09-21T16:58:02Z https://www.ideals.illinois.edu/oai-pmh

oai:www.ideals.illinois.edu:2142/107893 2023-07-11 col_2142_5131 col_2142_16340 com_2142_5130 com_2142_16339 com_2142_8903

Erdoğan, Burak Tzirakis, Nikolaos Li, Xiaochun Albin, Pierre Harris, Terence L. J. 2020-08-26T21:54:21Z 2020-08-26T21:54:21Z 2020-04-21 2020-05 In the first part of this thesis, it is shown that if $A \subseteq \mathbb{R}^3$ is a Borel set of Hausdorff dimension $\dim A > 3/2$, then for a.e.~$\theta \in [0,2\pi)$ the projection $\pi_{\theta}(A)$ of $A$ onto the 2-dimensional plane orthogonal to $\frac{1}{\sqrt{2}}(\cos \theta, \sin \theta, 1)$ satisfies \[ \dim \pi_{\theta}(A) \geq \min\left\{\frac{4\dim A}{9} + \frac{5}{6},2 \right\}. \] This improves the bound of Oberlin and Oberlin \cite{oberlin}, and of Orponen and Venieri \cite{venieri}, for $\dim A \in (1.5,2.4)$. In the second part, an improved lower bound is given for the decay of conical averages of Fourier transforms of measures, for cones of dimension $d \geq 4$. The proof uses a weighted version of the broad restriction inequality, a narrow decoupling inequality for the cone, and some techniques of Du and Zhang \cite{zhang} originally developed for the Schrödinger equation. Most of the work in this thesis was published by the author in different forms in \cite{THarris1} and \cite{THarris3}. Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2020-08-25 without embargo terms The student, Terence Harris, accepted the attached license on 2020-04-17 at 12:08. The student, Terence Harris, submitted this Dissertation for approval on 2020-04-17 at 12:19. This Dissertation was approved for publication on 2020-04-21 at 10:15. DSpace SAF Submission Ingestion Package generated from Vireo submission #14994 on 2020-08-25 at 17:07:13 Made available in DSpace on 2020-08-26T21:54:21Z (GMT). No. of bitstreams: 9 HARRIS-DISSERTATION-2020.pdf: 661822 bytes, checksum: c8391d66d95a7720fa6546506d16fb9a (MD5) R3proj.tex: 99937 bytes, checksum: d78ae595627898ae33cd7ce0535b944d (MD5) abs.tex: 1040 bytes, checksum: c899b4e40289e243a2b9028f1115c734 (MD5) ack.tex: 1179 bytes, checksum: 36933121aa2b24a8678ed221f09dcc08 (MD5) decay.tex: 56777 bytes, checksum: 06d3315edc67bfbbbff86cba562e5639 (MD5) intro.tex: 16083 bytes, checksum: 98230bdbe5e1849c89c76faf9857b139 (MD5) thesis.tex: 5632 bytes, checksum: 3361d76470d4b44df4989020c48eca60 (MD5) thesisrefs.bib: 84735 bytes, checksum: d508ef87ade4ffb043e3834a55c01f66 (MD5) LICENSE.txt: 4211 bytes, checksum: 0dfac0c4af48135a8d3393e283bf2cd8 (MD5) Previous issue date: 2020-04-21 application/pdf http://hdl.handle.net/2142/107893 en Copyright 2020 Terence Harris Hausdorff dimension Orthogonal projections Restricted projection families and weighted Fourier restriction text Thesis Mathematics Mathematics University of Illinois at Urbana-Champaign Dissertation Ph.D.