dc.contributor.advisor | Iosevich, Alex, 1967- | eng |
dc.contributor.author | Chapman, Jeremy Michael, 1982- | eng |
dc.date.issued | 2010 | eng |
dc.date.submitted | 2010 Spring | eng |
dc.description | Title from PDF of title page (University of Missouri--Columbia, viewed on May 24, 2010). | eng |
dc.description | The entire thesis text is included in the research.pdf file; the official abstract appears in the short.pdf file; a non-technical public abstract appears in the public.pdf file. | eng |
dc.description | Dissertation advisor: Dr. Alex Iosevich. | eng |
dc.description | Vita. | eng |
dc.description | Includes bibliographical references. | eng |
dc.description | Ph. D. University of Missouri--Columbia 2010. | eng |
dc.description | Dissertations, Academic -- University of Missouri--Columbia -- Mathematics. | eng |
dc.description.abstract | We study a variety of combinatorial distance and dot product related problems in vector spaces over finite fields. First, we focus on the generation of the Special Linear Group whose elements belong to a finite field with q elements. Given A [subset of] Fq, we use Fourier analytic methods to determine how large A needs to be to ensure that a certain product set contains a positive proportion of all the elements of SL₂(Fq). We also study a variety of distance and dot product sets related to the Erd̋os-Falconer distance problem. In general, the Erd̋os-Falconer distance problem asks for the number of distances determined by a set of points. The classical Erdős distance problem asks for the minimal number of distinct distances determined by a finite point set in Rd, where d [is reducible to] 2. The Falconer distance problem, which is the continuous analog of the Erd̋os distance problem, asks to find s₀ [greater than] 0 such that if the Hausdorff dimension of E is greater than s₀, then the Lebesgue measure of [symmetric difference] (E) is positive. A generalization of the Erdős-Falconer distance problem in vector spaces over finite fields is to determine the minimal [alpha] [greater than] 0 such that E contains a congruent copy of every k dimensional simplex whenever [E] [almost equal to] q [alpha]. We improve on known results (for k [greater than] 3) using Fourier analytic methods, showing that [alpha] may be taken to be d+k2 . If E is a subset of a sphere, then we get a stronger result which shows that [alpha] may be taken to be d+k -1 [over] 2. | eng |
dc.format.extent | iv, 42 pages | eng |
dc.identifier.merlin | b77789787 | eng |
dc.identifier.merlin | b77789787 | eng |
dc.identifier.oclc | 656841297 | eng |
dc.identifier.uri | https://hdl.handle.net/10355/8285 | |
dc.identifier.uri | https://doi.org/10.32469/10355/8285 | eng |
dc.language | English | eng |
dc.publisher | University of Missouri--Columbia | eng |
dc.relation.ispartof | 2010 Freely available dissertations (MU) | eng |
dc.relation.ispartofcommunity | University of Missouri-Columbia. Graduate School. Theses and Dissertations. Dissertations. 2010 Dissertations | eng |
dc.subject.lcsh | Erdős, Paul -- 1913-1996 | eng |
dc.subject.lcsh | Falconer, K. J., 1952- | eng |
dc.subject.lcsh | Finite geometries | eng |
dc.subject.lcsh | Combinatorial analysis | eng |
dc.subject.lcsh | Vector spaces | eng |
dc.title | Finite point configurations and projection theorems in vector spaces over finite fields | eng |
dc.type | Thesis | eng |
thesis.degree.discipline | Mathematics (MU) | eng |
thesis.degree.grantor | University of Missouri--Columbia | eng |
thesis.degree.level | Doctoral | eng |
thesis.degree.name | Ph. D. | eng |