Zeev Dvir

On the size of Kakeya sets in finite fields

A Kakeya set is a subset of F^n, where F is a finite field of q elements, that contains a line in every direction. In this paper we show that the size of every Kakeya set is at least C_n * q^n, where C_n depends only on n. This improves the previously best lower bound for general n of ~q^{4n/7}.

Locally Decodable Codes with Two Queries and Polynomial Identity Testing for Depth 3 Circuits

In this work we study two, seemingly unrelated, notions. Locally decodable codes (LDCs) are codes that allow the recovery of each message bit from a constant number of entries of the codeword. Polynomial identity testing (PIT) is one of the fundamental problems of algebraic complexity: we are given a circuit computing a multivariate polynomial and we have to determine whether the polynomial is identically zero. We improve known results on LDCs and on polynomial identity testing and show a relation between the two notions. In particular we obtain the following results: (1) We show that if

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

E: \mathbb{F}^n \mapsto \mathbb{F}^m

Locally decodable codes with 2 queries and polynomial identity testing for depth 3 circuits

is a linear LDC with two queries, then

Hardness-Randomness Tradeoffs for Bounded Depth Arithmetic Circuits

m = \exp(\Omega(n))

Extractors And Rank Extractors For Polynomial Sources

. Previously this was known only for fields of size

Subspace evasive sets

\ll 2^n

Matching Vector Codes

[O. Goldreich e, Comput. Complexity, 15 (2006), pp. 263-296]. (2) We show that from every depth 3 arithmetic circuit (

Rank bounds for design matrices with applications to combinatorial geometry and locally correctable codes

\Sigma\Pi\Sigma

Extractors for Varieties

circuit),