{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/107893"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/107893","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Restricted projection families and weighted Fourier restriction","abstract":"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}.","abstract_html":"In the first part of this thesis, it is shown that if <span class=\"etd-inline-math\">A \\subseteq \\mathbb{R}<sup>3</sup></span> is a Borel set of Hausdorff dimension $\\dim A &gt; 3/2$, then for a.e.~<span class=\"etd-inline-math\">&theta; \\in [0,2&pi;)</span> the projection <span class=\"etd-inline-math\">&pi;<sub>&theta;</sub>(A)</span> of $A$ onto the 2-dimensional plane orthogonal to <span class=\"etd-inline-math\">\\frac{1}{\\sqrt{2}}(\\cos &theta;, \\sin &theta;, 1)</span> 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}.","abstract_has_math":true,"creators":["Harris, Terence L. J."],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Erdoğan, Burak","Tzirakis, Nikolaos","Li, Xiaochun","Albin, Pierre"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2020,"date_issued":"2020-08-26T21:54:21Z","date_published":"2020-08-26T21:54:21Z","updated_at":"2026-07-22T22:24:47Z","subjects":["Hausdorff dimension","Orthogonal projections"],"languages":["en"],"rights":["Copyright 2020 Terence Harris"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/107893","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Erdoğan, Burak","Tzirakis, Nikolaos","Li, Xiaochun","Albin, Pierre"]},{"key":"dc:creator","label":"Author","values":["Harris, Terence L. J."]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2020-08-26T21:54:21Z","2020-04-21","2020-05"]},{"key":"dc:type","label":"Dc Type","values":["text","Thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Hausdorff dimension","Orthogonal projections"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2020 Terence Harris"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/107893"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["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). 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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). 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