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dc.contributor.advisorIosevich, Alex, 1967-eng
dc.contributor.authorChapman, Jeremy Michael, 1982-eng
dc.date.issued2010eng
dc.date.submitted2010 Springeng
dc.descriptionTitle from PDF of title page (University of Missouri--Columbia, viewed on May 24, 2010).eng
dc.descriptionThe 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.descriptionDissertation advisor: Dr. Alex Iosevich.eng
dc.descriptionVita.eng
dc.descriptionIncludes bibliographical references.eng
dc.descriptionPh. D. University of Missouri--Columbia 2010.eng
dc.descriptionDissertations, Academic -- University of Missouri--Columbia -- Mathematics.eng
dc.description.abstractWe 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.extentiv, 42 pageseng
dc.identifier.merlinb77789787eng
dc.identifier.merlinb77789787eng
dc.identifier.oclc656841297eng
dc.identifier.urihttps://hdl.handle.net/10355/8285
dc.identifier.urihttps://doi.org/10.32469/10355/8285eng
dc.languageEnglisheng
dc.publisherUniversity of Missouri--Columbiaeng
dc.relation.ispartof2010 Freely available dissertations (MU)eng
dc.relation.ispartofcommunityUniversity of Missouri-Columbia. Graduate School. Theses and Dissertations. Dissertations. 2010 Dissertationseng
dc.subject.lcshErdős, Paul -- 1913-1996eng
dc.subject.lcshFalconer, K. J., 1952-eng
dc.subject.lcshFinite geometrieseng
dc.subject.lcshCombinatorial analysiseng
dc.subject.lcshVector spaceseng
dc.titleFinite point configurations and projection theorems in vector spaces over finite fieldseng
dc.typeThesiseng
thesis.degree.disciplineMathematics (MU)eng
thesis.degree.grantorUniversity of Missouri--Columbiaeng
thesis.degree.levelDoctoraleng
thesis.degree.namePh. D.eng


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