{"id":{"repo_id":"washington","oai_identifier":"oai:digital.lib.washington.edu:1773/54048"},"canonical_url":"https://search.dev.ndltd.org/etd/washington/oai:digital.lib.washington.edu:1773/54048","repository":{"repo_id":"washington","name":"University of Washington","base_url":"https://digital.lib.washington.edu/server/oai/request"},"display":{"title":"Higher-Dimensional Analogues of Two Theorems of Orponen on Exceptional Sets","abstract":"Let $A \\subseteq \\mathbb{R}^n$ be a Borel set of Hausdorff dimension $\\dim_{\\mathrm H} A = s$ and, for each $m$-plane $V$ in the Grassmannian $\\mathbf{Gr}(n,m)$, let $\\pi_V : \\mathbb{R}^n \\to V$ be the orthogonal projection of $\\mathbb{R}^n$ onto $V$. \\textit{Marstrand's projection theorem} states that $\\dim_{\\mathrm H} \\pi_V(A) = \\min \\, \\{s,m\\}$ for almost every $V \\in \\mathbf{Gr}(n,m)$, and \\textit{Marstrand's slicing theorem} states that the set of $\\mathbf{x} \\in V$ such that $\\dim_{\\mathrm H} \\big( \\pi_V^{-1}(\\mathbf{x}) \\cap A \\big) = s-m$ has positive Lebesgue measure for almost every $V \\in \\mathbf{Gr}(n,m)$ provided $s > m$. Informally, for all but a few subspaces $V$, the shadow of $A$ on $V$ has the largest possible dimension, and there are many sections of $A$ orthogonal to $V$ of essentially maximal dimension. In this dissertation, we prove two results concerning the sizes of \\textit{exceptional sets} of projections and slices—the null sets of parameters $V$ for which the conclusion of Marstrand's projection or slicing theorem fails, respectively. These are, in particular, higher-dimensional analogues of two theorems of Tuomas Orponen. The first result states that, if linear maps are \\textit{almost dimension conserving} for $A$, then the exceptional sets of orthogonal projections of $A$ have small packing dimension. This applies, for example, to (weakly) Furstenberg homogeneous sets and to certain self-similar and graph-directed sets. The second concerns slices by fibers of the \\textit{generalized projections} of Peres and Schlag. Roughly, the conclusion of Marstrand's slicing theorem holds for very general families of nonlinear projections, and the exceptional sets of slices have small Hausdorff dimension. In fact, not only is the set of exceptional slices of $A$ small, but the union of the exceptional sets of all positive-$\\mathcal{H}^s$-measure Borel \\textit{subsets} of $A$ is small.","abstract_html":"Let <span class=\"etd-inline-math\">A \\subseteq \\mathbb{R}<sup>n</sup></span> be a Borel set of Hausdorff dimension <span class=\"etd-inline-math\">\\dim<sub>\\mathrm H</sub> A = s</span> and, for each $m$-plane $V$ in the Grassmannian <span class=\"etd-inline-math\"><strong>Gr</strong>(n,m)</span>, let <span class=\"etd-inline-math\">&pi;<sub>V</sub> : \\mathbb{R}<sup>n</sup> \\to V</span> be the orthogonal projection of <span class=\"etd-inline-math\">\\mathbb{R}<sup>n</sup></span> onto $V$. \\textit{Marstrand&#x27;s projection theorem} states that <span class=\"etd-inline-math\">\\dim<sub>\\mathrm H</sub> &pi;<sub>V</sub>(A) = \\min   \\{s,m\\}</span> for almost every <span class=\"etd-inline-math\">V \\in <strong>Gr</strong>(n,m)</span>, and \\textit{Marstrand&#x27;s slicing theorem} states that the set of <span class=\"etd-inline-math\"><strong>x</strong> \\in V</span> such that <span class=\"etd-inline-math\">\\dim<sub>\\mathrm H</sub> \\big( &pi;<sub>V</sub><sup>-1</sup>(<strong>x</strong>) \\cap A \\big) = s-m</span> has positive Lebesgue measure for almost every <span class=\"etd-inline-math\">V \\in <strong>Gr</strong>(n,m)</span> provided $s &gt; m$. Informally, for all but a few subspaces $V$, the shadow of $A$ on $V$ has the largest possible dimension, and there are many sections of $A$ orthogonal to $V$ of essentially maximal dimension. In this dissertation, we prove two results concerning the sizes of \\textit{exceptional sets} of projections and slices—the null sets of parameters $V$ for which the conclusion of Marstrand&#x27;s projection or slicing theorem fails, respectively. These are, in particular, higher-dimensional analogues of two theorems of Tuomas Orponen. The first result states that, if linear maps are \\textit{almost dimension conserving} for $A$, then the exceptional sets of orthogonal projections of $A$ have small packing dimension. This applies, for example, to (weakly) Furstenberg homogeneous sets and to certain self-similar and graph-directed sets. The second concerns slices by fibers of the \\textit{generalized projections} of Peres and Schlag. Roughly, the conclusion of Marstrand&#x27;s slicing theorem holds for very general families of nonlinear projections, and the exceptional sets of slices have small Hausdorff dimension. In fact, not only is the set of exceptional slices of $A$ small, but the union of the exceptional sets of all positive-<span class=\"etd-inline-math\">\\mathcal{H}<sup>s</sup></span>-measure Borel \\textit{subsets} of $A$ is small.","abstract_has_math":true,"creators":["Bushling, Ryan Edward"],"institution":null,"degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":["Wilson, Bobby L"],"committee_chairs":[],"committee_members":[],"year":2025,"date_issued":"2025-10-02","date_published":"2025-10-02","updated_at":"2026-07-24T05:58:14Z","subjects":["Exceptional set","Fractal dimension","Projection","Mathematics"],"languages":["en_US"],"rights":["CC BY"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/1773/54048","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Wilson, Bobby L"]},{"key":"dc:creator","label":"Author","values":["Bushling, Ryan Edward"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2025-10-02T16:11:30Z"]},{"key":"dc:date.issued","label":"Date","values":["2025-10-02"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Exceptional set","Fractal dimension","Projection","Mathematics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["en_US"]},{"key":"dc:rights","label":"Dc Rights","values":["CC BY"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.other","label":"Dc Identifier Other","values":["Bushling_washington_0250E_28782.pdf"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://hdl.handle.net/1773/54048"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Thesis (Ph.D.)--University of Washington, 2025"]},{"key":"dc:description.abstract","label":"Abstract","values":["Let $A \\subseteq \\mathbb{R}^n$ be a Borel set of Hausdorff dimension $\\dim_{\\mathrm H} A = s$ and, for each $m$-plane $V$ in the Grassmannian $\\mathbf{Gr}(n,m)$, let $\\pi_V : \\mathbb{R}^n \\to V$ be the orthogonal projection of $\\mathbb{R}^n$ onto $V$. \\textit{Marstrand's projection theorem} states that $\\dim_{\\mathrm H} \\pi_V(A) = \\min \\, \\{s,m\\}$ for almost every $V \\in \\mathbf{Gr}(n,m)$, and \\textit{Marstrand's slicing theorem} states that the set of $\\mathbf{x} \\in V$ such that $\\dim_{\\mathrm H} \\big( \\pi_V^{-1}(\\mathbf{x}) \\cap A \\big) = s-m$ has positive Lebesgue measure for almost every $V \\in \\mathbf{Gr}(n,m)$ provided $s > m$. Informally, for all but a few subspaces $V$, the shadow of $A$ on $V$ has the largest possible dimension, and there are many sections of $A$ orthogonal to $V$ of essentially maximal dimension. In this dissertation, we prove two results concerning the sizes of \\textit{exceptional sets} of projections and slices—the null sets of parameters $V$ for which the conclusion of Marstrand's projection or slicing theorem fails, respectively. These are, in particular, higher-dimensional analogues of two theorems of Tuomas Orponen. The first result states that, if linear maps are \\textit{almost dimension conserving} for $A$, then the exceptional sets of orthogonal projections of $A$ have small packing dimension. This applies, for example, to (weakly) Furstenberg homogeneous sets and to certain self-similar and graph-directed sets. The second concerns slices by fibers of the \\textit{generalized projections} of Peres and Schlag. Roughly, the conclusion of Marstrand's slicing theorem holds for very general families of nonlinear projections, and the exceptional sets of slices have small Hausdorff dimension. In fact, not only is the set of exceptional slices of $A$ small, but the union of the exceptional sets of all positive-$\\mathcal{H}^s$-measure Borel \\textit{subsets} of $A$ is small."]},{"key":"dc:format.mimetype","label":"Dc Format Mimetype","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Higher-Dimensional Analogues of Two Theorems of Orponen on Exceptional Sets"]}]}],"canonical_facts":{"dc:contributor.advisor":["Wilson, Bobby L"],"dc:creator":["Bushling, Ryan Edward"],"dc:date.accessioned":["2025-10-02T16:11:30Z"],"dc:date.issued":["2025-10-02"],"dc:description":["Thesis (Ph.D.)--University of Washington, 2025"],"dc:description.abstract":["Let $A \\subseteq \\mathbb{R}^n$ be a Borel set of Hausdorff dimension $\\dim_{\\mathrm H} A = s$ and, for each $m$-plane $V$ in the Grassmannian $\\mathbf{Gr}(n,m)$, let $\\pi_V : \\mathbb{R}^n \\to V$ be the orthogonal projection of $\\mathbb{R}^n$ onto $V$. \\textit{Marstrand's projection theorem} states that $\\dim_{\\mathrm H} \\pi_V(A) = \\min \\, \\{s,m\\}$ for almost every $V \\in \\mathbf{Gr}(n,m)$, and \\textit{Marstrand's slicing theorem} states that the set of $\\mathbf{x} \\in V$ such that $\\dim_{\\mathrm H} \\big( \\pi_V^{-1}(\\mathbf{x}) \\cap A \\big) = s-m$ has positive Lebesgue measure for almost every $V \\in \\mathbf{Gr}(n,m)$ provided $s > m$. Informally, for all but a few subspaces $V$, the shadow of $A$ on $V$ has the largest possible dimension, and there are many sections of $A$ orthogonal to $V$ of essentially maximal dimension. In this dissertation, we prove two results concerning the sizes of \\textit{exceptional sets} of projections and slices—the null sets of parameters $V$ for which the conclusion of Marstrand's projection or slicing theorem fails, respectively. These are, in particular, higher-dimensional analogues of two theorems of Tuomas Orponen. The first result states that, if linear maps are \\textit{almost dimension conserving} for $A$, then the exceptional sets of orthogonal projections of $A$ have small packing dimension. This applies, for example, to (weakly) Furstenberg homogeneous sets and to certain self-similar and graph-directed sets. The second concerns slices by fibers of the \\textit{generalized projections} of Peres and Schlag. Roughly, the conclusion of Marstrand's slicing theorem holds for very general families of nonlinear projections, and the exceptional sets of slices have small Hausdorff dimension. In fact, not only is the set of exceptional slices of $A$ small, but the union of the exceptional sets of all positive-$\\mathcal{H}^s$-measure Borel \\textit{subsets} of $A$ is small."],"dc:format.mimetype":["application/pdf"],"dc:identifier.other":["Bushling_washington_0250E_28782.pdf"],"dc:identifier.uri":["https://hdl.handle.net/1773/54048"],"dc:language.iso":["en_US"],"dc:rights":["CC BY"],"dc:subject":["Exceptional set","Fractal dimension","Projection","Mathematics"],"dc:title":["Higher-Dimensional Analogues of Two Theorems of Orponen on Exceptional Sets"],"dc:type":["Thesis"]},"updated_at":"2026-07-24T05:58:14Z"}