78 research outputs found

    Extensions to the Method of Multiplicities, with applications to Kakeya Sets and Mergers

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    We extend the "method of multiplicities" to get the following results, of interest in combinatorics and randomness extraction. (A) We show that every Kakeya set (a set of points that contains a line in every direction) in \F_q^n must be of size at least qn/2nq^n/2^n. This bound is tight to within a 2+o(1)2 + o(1) factor for every nn as qq \to \infty, compared to previous bounds that were off by exponential factors in nn. (B) We give improved randomness extractors and "randomness mergers". Mergers are seeded functions that take as input Λ\Lambda (possibly correlated) random variables in {0,1}N\{0,1\}^N and a short random seed and output a single random variable in {0,1}N\{0,1\}^N that is statistically close to having entropy (1δ)N(1-\delta) \cdot N when one of the Λ\Lambda input variables is distributed uniformly. The seed we require is only (1/δ)logΛ(1/\delta)\cdot \log \Lambda-bits long, which significantly improves upon previous construction of mergers. (C) Using our new mergers, we show how to construct randomness extractors that use logarithmic length seeds while extracting 1o(1)1 - o(1) fraction of the min-entropy of the source. The "method of multiplicities", as used in prior work, analyzed subsets of vector spaces over finite fields by constructing somewhat low degree interpolating polynomials that vanish on every point in the subset {\em with high multiplicity}. The typical use of this method involved showing that the interpolating polynomial also vanished on some points outside the subset, and then used simple bounds on the number of zeroes to complete the analysis. Our augmentation to this technique is that we prove, under appropriate conditions, that the interpolating polynomial vanishes {\em with high multiplicity} outside the set. This novelty leads to significantly tighter analyses.Comment: 26 pages, now includes extractors with sublinear entropy los

    Kakeya sets and the method of multiplicities

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    Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2009.Cataloged from PDF version of thesis.Includes bibliographical references (p. 51-53).We extend the "method of multiplicities" to get the following results, of interest in combinatorics and randomness extraction. 1. We show that every Kakeya set (a set of points that contains a line in every direction) in F' must be of size at least qn/2n. This bound is tight to within a 2 + o(1) factor for every n as q -- oc, compared to previous bounds that were off by exponential factors in n. 2. We give improved randomness extractors and "randomness mergers". Mergers are seeded functions that take as input A (possibly correlated) random variables in {0, 1}N and a short random seed and output a single random variable in {0, 1}N that is statistically close to having entropy (1 - 6) - N when one of the A input variables is distributed uniformly. The seed we require is only (1/6) - log A-bits long, which significantly improves upon previous construction of mergers. 3. Using our new mergers, we show how to construct randomness extractors that use logarithmic length seeds while extracting 1- o(1) fraction of the min-entropy of the source. The "method of multiplicities", as used in prior work, analyzed subsets of vector spaces over finite fields by constructing somewhat low degree interpolating polynomials that vanish on every point in the subset with high multiplicity. The typical use of this method involved showing that the interpolating polynomial also vanished on some points outside the subset, and then used simple bounds on the number of zeroes to complete the analysis. Our augmentation to this technique is that we prove, under appropriate conditions, that the interpolating polynomial vanishes with high multiplicity outside the set. This novelty leads to significantly tighter analyses.by Shubhangi Saraf.S.M

    The method of multiplicities

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    Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2011.Cataloged from PDF version of thesis.Includes bibliographical references (p. 93-98).Polynomials have played a fundamental role in the construction of objects with interesting combinatorial properties, such as error correcting codes, pseudorandom generators and randomness extractors. Somewhat strikingly, polynomials have also been found to be a powerful tool in the analysis of combinatorial parameters of objects that have some algebraic structure. This method of analysis has found applications in works on list-decoding of error correcting codes, constructions of randomness extractors, and in obtaining strong bounds for the size of Kakeya Sets. Remarkably, all these applications have relied on very simple and elementary properties of polynomials such as the sparsity of the zero sets of low degree polynomials. In this thesis we improve on several of the results mentioned above by a more powerful application of polynomials that takes into account the information contained in the derivatives of the polynomials. We call this technique the method of multiplicities. The derivative polynomials encode information about the high multiplicity zeroes of the original polynomial, and by taking into account this information, we are about to meaningfully reason about the zero sets of polynomials of degree much higher than the underlying field size. This freedom of using high degree polynomials allows us to obtain new and improved constructions of error correcting codes, and qualitatively improved analyses of Kakeya sets and randomness extractors.by Shubhangi Saraf.Ph.D

    A finite version of the Kakeya problem

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    Let LL be a set of lines of an affine space over a field and let SS be a set of points with the property that every line of LL is incident with at least NN points of SS. Let DD be the set of directions of the lines of LL considered as points of the projective space at infinity. We give a geometric construction of a set of lines LL, where DD contains an Nn1N^{n-1} grid and where SS has size 2((1/2)N)n2((1/2)N)^n, given a starting configuration in the plane. We provide examples of such starting configurations for the reals and for finite fields. Following Dvir's proof of the finite field Kakeya conjecture and the idea of using multiplicities of Dvir, Kopparty, Saraf and Sudan, we prove a lower bound on the size of SS dependent on the ideal generated by the homogeneous polynomials vanishing on DD. This bound is maximised as ((1/2)N)n((1/2)N)^n plus smaller order terms, for n4n\geqslant 4, when DD contains the points of a Nn1N^{n-1} grid.Comment: A few minor changes to previous versio

    Counting joints in vector spaces over arbitrary fields

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    We give a proof of the "folklore" theorem that the Kaplan--Sharir--Shustin/Quilodr\'an result on counting joints associated to a family of lines holds in vector spaces over arbitrary fields, not just the reals. We also discuss a distributional estimate on the multiplicities of the joints in the case that the family of lines is sufficiently generic.Comment: Not intended for publication. References added and other minor edits in this versio

    Linear Hashing with \ell_\infty guarantees and two-sided Kakeya bounds

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    We show that a randomly chosen linear map over a finite field gives a good hash function in the \ell_\infty sense. More concretely, consider a set SFqnS \subset \mathbb{F}_q^n and a randomly chosen linear map L:FqnFqtL : \mathbb{F}_q^n \to \mathbb{F}_q^t with qtq^t taken to be sufficiently smaller than S|S|. Let USU_S denote a random variable distributed uniformly on SS. Our main theorem shows that, with high probability over the choice of LL, the random variable L(US)L(U_S) is close to uniform in the \ell_\infty norm. In other words, every element in the range Fqt\mathbb{F}_q^t has about the same number of elements in SS mapped to it. This complements the widely-used Leftover Hash Lemma (LHL) which proves the analog statement under the statistical, or 1\ell_1, distance (for a richer class of functions) as well as prior work on the expected largest 'bucket size' in linear hash functions [ADMPT99]. Our proof leverages a connection between linear hashing and the finite field Kakeya problem and extends some of the tools developed in this area, in particular the polynomial method
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