Finite point configurations and projection theorems in vector spaces over finite fields

Title from PDF of title page (University of Missouri--Columbia, viewed on May 24, 2010).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.Dissertation advisor: Dr. Alex Iosevich.Vita.Includes bibliographical references.Ph. D. University of Missouri--Columbia 2010.Dissertations, Academic -- University of Missouri--Columbia -- Mathematics.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

Nontrivial t-Designs over Finite Fields Exist for All t

A $t$-$(n,k,\lambda)$ design over \F_q is a collection of $k$-dimensional subspaces of \F_q^n, called blocks, such that each $t$-dimensional subspace of \F_q^n is contained in exactly $\lambda$ blocks. Such $t$-designs over \F_q are the $q$-analogs of conventional combinatorial designs. Nontrivial $t$-$(n,k,\lambda)$ designs over \F_q are currently known to exist only for $t \leq 3$. Herein, we prove that simple (meaning, without repeated blocks) nontrivial $t$-$(n,k,\lambda)$ designs over \F_q exist for all $t$ and $q$, provided that $k > 12t$ and $n$ is sufficiently large. This may be regarded as a $q$-analog of the celebrated Teirlinck theorem for combinatorial designs

Random Sampling in Computational Algebra: Helly Numbers and Violator Spaces

This paper transfers a randomized algorithm, originally used in geometric optimization, to computational problems in commutative algebra. We show that Clarkson's sampling algorithm can be applied to two problems in computational algebra: solving large-scale polynomial systems and finding small generating sets of graded ideals. The cornerstone of our work is showing that the theory of violator spaces of G\"artner et al.\ applies to polynomial ideal problems. To show this, one utilizes a Helly-type result for algebraic varieties. The resulting algorithms have expected runtime linear in the number of input polynomials, making the ideas interesting for handling systems with very large numbers of polynomials, but whose rank in the vector space of polynomials is small (e.g., when the number of variables and degree is constant).Comment: Minor edits, added two references; results unchange

Problems on q-Analogs in Coding Theory

The interest in $q$-analogs of codes and designs has been increased in the last few years as a consequence of their new application in error-correction for random network coding. There are many interesting theoretical, algebraic, and combinatorial coding problems concerning these q-analogs which remained unsolved. The first goal of this paper is to make a short summary of the large amount of research which was done in the area mainly in the last few years and to provide most of the relevant references. The second goal of this paper is to present one hundred open questions and problems for future research, whose solution will advance the knowledge in this area. The third goal of this paper is to present and start some directions in solving some of these problems.Comment: arXiv admin note: text overlap with arXiv:0805.3528 by other author

The Tensor Track, III

We provide an informal up-to-date review of the tensor track approach to quantum gravity. In a long introduction we describe in simple terms the motivations for this approach. Then the many recent advances are summarized, with emphasis on some points (Gromov-Hausdorff limit, Loop vertex expansion, Osterwalder-Schrader positivity...) which, while important for the tensor track program, are not detailed in the usual quantum gravity literature. We list open questions in the conclusion and provide a rather extended bibliography.Comment: 53 pages, 6 figure