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

Abstract

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 <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> must be of size at least q <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> /2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> . This bound is tight to within a 2 + o(1) factor for every n as q ? ?, compared to previous bounds that were off by exponential factors in n. 2) We give an improved construction of "randomness mergers". Mergers are seeded functions that take as input ? (possibly correlated) random variables in {0,1} <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</sup> and a short random seed, and output a single random variable in {0,1} <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</sup> that is statistically close to having entropy (1 - ?) ? N when one of the ? input variables is distributed uniformly. The seed we require is only (1/?) ? log ?-bits long, which significantly improves upon previous construction of mergers. 3) 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. Previous results could extract only a constant fraction of the entropy while maintaining logarithmic seed length. 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. To develop the extended method of multiplicities we provide a number of basic technical results about multiplicity of zeroes of polynomials that may be of general use. For instance, we strengthen the Schwartz-Zippel lemma to show that the expected multiplicity of zeroes of a non-zero degree d polynomial at a random point in S <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> , for any finite subset S of the underlying field, is at most d/|S|.