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 in F n q, the n-dimensional vector space over the finite field on q elements, must be of size at least q n /2 n. This bound is tight to within a 2 + o(1) factor for every n as q → ∞. 2. We give improved “randomness mergers”, i.e., seeded functions that take as input k (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−δ)·N when one of the k input variables is distributed uniformly. The seed we require is only (1/δ)·log k-bits long, which significantly improves upon previous construction of mergers. 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

In a recent breakthrough, Dvir showed that every Kakeya set in F n must be of cardinality at least cn|F | n where cn ≈ 1/n!. We improve this lower bound to β n |F | n for a constant β> 0. This pins down the growth of the leading constant to the right form as a function of n. Let F be a finite field of q elements. Definition 1 (Kakeya Set) A set K ⊆ F n is said to be a Kakeya set in F n, if for every b ∈ F n, there exists a point a ∈ F n such that for every t ∈ F, the point a + t · b ∈ K. We show: Theorem 2 There exist constants c0, c1> 0 such that for all n, if K is a Kakeya set in F n then |K | ≥ c0 · (c1 · q) n. Remark Our proofs give some tradeoffs on the constants c0, c1 that are achievable. We comment on the constants at the end of the paper. The question of establishing lower bounds on the size of Kakeya sets was posed in Wolff [7]. Till recently, the best known lower bound on the size of Kakeya sets was of the form q αn for some α < 1. In a recent breakthrough Dvir [1] showed that every Kakeya set must have cardinality at least cnq n for cn = (n!) −1.

Abstract not found

Abstract. We prove the endpoint case of the multilinear Kakeya conjecture of Bennett, Carbery, and Tao. The proof uses the polynomial method introduced by Dvir. In [1], Bennett, Carbery, and Tao formulated a multilinear Kakeya conjecture, and they proved the conjecture except for the endpoint case. In this paper, we slightly sharpen their result by proving the endpoint case of the conjecture. Our method of proof is very different from the proof of Bennett, Carbery, and Tao. The original proof was based on monotonicity estimates for heat flows. In 2007, Dvir [2] made a breakthrough on the Kakeya problem, proving the Kakeya conjecture over finite fields. His proof used polynomials in a crucial way. It was not clear whether Dvir’s approach could be adapted to prove estimates in Euclidean space. Our proof of the multilinear Kakeya conjecture is based on Dvir’s polynomial method. In my opinion, the method of proof is as interesting as the result. The multilinear Kakeya conjecture concerns the overlap properties of cylindrical tubes in R n. Roughly, the (multilinear) Kakeya conjecture says that cylinders pointing in different directions cannot overlap too much. Before coming to the Bennett-Carbery-Tao multilinear estimate, I want to state a weaker result, because it’s easier to understand and easier to prove. To be clear about the notation, a cylinder of radius R around a line L ⊂ R n is the set of all points x ∈ R n within a distance R of the line L. We call the line L the core of the cylinder. Theorem 1. Suppose we have a finite collection of cylinders Tj,a ⊂ R n, where 1 ≤ j ≤ n, and 1 ≤ a ≤ A for some integer A. Each cylinder has radius 1. Moreover, each cylinder Tj,a runs nearly parallel to the xj-axis. More precisely, we assume that the angle between the core of Tj,a and the xj-axis is at most (100n) −1. We let I be the set of points that belong to at least one cylinder in each direction. In symbols, Then V ol(I) ≤ C(n)A n n−1.

Additive combinatorics is the branch of combinatorics where the objects of study are subsets of the integers or of other abelian groups, and one is interested in properties and patterns that can be expressed in terms of linear equations. More generally, arithmetic combinatorics deals with properties and patterns that can be expressed via additions and multiplications. In the past ten years, additive and arithmetic combinatorics have been extremely successful areas of mathematics, featuring a convergence of techniques from graph theory, analysis and ergodic theory. They have helped prove long-standing open questions in additive number theory, and they offer much promise of future progress. Techniques from additive and arithmetic combinatorics have found several applications in computer science too, to property testing, pseudorandomness, PCP constructions, lower bounds, and extractor constructions. Typically, whenever a technique from additive or arithmetic combinatorics becomes understood by computer scientists, it finds some application. Considering that there is still a lot of additive and arithmetic combinatorics that computer scientists do not understand (and, the field being very active, even more will be developed in the near future), there seems to be much potential for future connections and applications. 1

Let L be a set of n lines in R d, for d ≥ 3. A joint of L is a point incident to at least d lines of L, not all in a common hyperplane. Using a very simple algebraic proof technique, we show that the maximum possible number of joints of L is Θ(n d/(d−1)). For d = 3, this is a considerable simplification of the orignal algebraic proof of Guth and Katz [9], and of the follow-up simpler proof of Elekes et al. [6]. Some extensions, e.g., to the case of joints of algebraic curves, are also presented.

Abstract. Error-correcting codes tackle the fundamental problem of recovering from errors during data communication and storage. A basic issue in coding theory concerns the modeling of the channel noise. Shannon’s theory models the channel as a stochastic process with a known probability law. Hamming suggested a combinatorial approach where the channel causes worst-case errors subject only to a limit on the number of errors. These two approaches share a lot of common tools, however in terms of quantitative results, the classical results for worst-case errors were much weaker. We survey recent progress on list decoding, highlighting its power and generality as an avenue to construct codes resilient to worst-case errors with information rates similar to what is possible against probabilistic errors. In particular, we discuss recent explicit constructions of list-decodable codes with information-theoretically optimal redundancy that is arbitrarily close to the fraction of symbols that can be corrupted by worst-case errors.

Randomness extractors are efficient algorithms which convert weak random sources into nearly perfect ones. While such purification of randomness was the original motivation for constructing extractors, these constructions turn out to have strong pseudorandom properties which found applications in diverse areas of computer science and combinatorics. We will highlight some of the applications, as well as recent constructions achieving near-optimal extraction.

Recently, additive combinatorics has blossomed into a vibrant area in mathematical sciences. But it seems to be a difficult area to define – perhaps because of a blend of ideas and techniques from several seemingly unrelated contexts which are used there. One might say that additive combinatorics is a branch of mathematics concerning the study of additive structures in sets equipped with a group structure – we may have other structure that interacts with this group structure. This newly emerging field has seen tremendous advances over the last few years, and has recently become a focus of attention among both mathematicians and computer scientists. This fascinating area has been enriched by its formidable links to combinatorics, number theory, harmonic analysis, ergodic theory, and some other branches; all deeply cross-fertilize each other, holding great promise for all of them! There is a considerable number of incredible problems, results, and novel applications in this thriving area. In this exposition, we attempt to provide an illuminating overview of some conspicuous breakthroughs in this captivating field, together with a number of seminal applications to sundry parts of mathematics and some other disciplines, with emphasis on computer science and cryptography.

To me, 2010 looks as annus mirabilis, a miraculous year, in several areas of my mathematical interests. Below I list seven highlights and breakthroughs, mostly in discrete geometry, hoping to share some of my wonder and pleasure with the readers. Of course, hardly any of these great results have come out of the blue: usually the paper I refer to adds the last step to earlier ideas. Since this is an extended abstract (of a nonexistent paper), I will be rather brief, or sometimes completely silent, about the history, with apologies to the unmentioned giants on whose shoulders the authors I do mention have been standing. 1 A careful reader may notice that together with these great results, I will also advertise some smaller results of mine. • Larry Guth and Nets Hawk Katz [16] completed a bold project of György Elekes (whose previous stage is reported in [10]) and obtained a neartight bound for the Erdős distinct distances problem: they proved that every n points in the plane determine at least Ω(n / log n) distinct distances. This almost matches the best known upper bound of O(n / √ √ √ log n), attained for the n × n grid. Their proof and some related results and methods constitute the main topic of this note, and will be discussed later. • János Pach and Gábor Tardos [27] found tight lower bounds for the size of ε-nets for geometric set systems. 2 It has been known for a long time