Subspace Evasive Sets

In this work we describe an explicit, simple, construction of large subsets of F^n, where F is a finite field, that have small intersection with every k-dimensional affine subspace. Interest in the explicit construction of such sets, termed subspace-evasive sets, started in the work of Pudlak and Rodl (2004) who showed how such constructions over the binary field can be used to construct explicit Ramsey graphs. More recently, Guruswami (2011) showed that, over large finite fields (of size polynomial in n), subspace evasive sets can be used to obtain explicit list-decodable codes with optimal rate and constant list-size. In this work we construct subspace evasive sets over large fields and use them to reduce the list size of folded Reed-Solomon codes form poly(n) to a constant.Comment: 16 page

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}.Comment: Improved bound and added reference

Improved rank bounds for design matrices and a new proof of Kelly's theorem

We study the rank of complex sparse matrices in which the supports of different columns have small intersections. The rank of these matrices, called design matrices, was the focus of a recent work by Barak et. al. (BDWY11) in which they were used to answer questions regarding point configurations. In this work we derive near-optimal rank bounds for these matrices and use them to obtain asymptotically tight bounds in many of the geometric applications. As a consequence of our improved analysis, we also obtain a new, linear algebraic, proof of Kelly's theorem, which is the complex analog of the Sylvester-Gallai theorem

Outlaw distributions and locally decodable codes

Locally decodable codes (LDCs) are error correcting codes that allow for decoding of a single message bit using a small number of queries to a corrupted encoding. Despite decades of study, the optimal trade-off between query complexity and codeword length is far from understood. In this work, we give a new characterization of LDCs using distributions over Boolean functions whose expectation is hard to approximate (in~$L_\infty$~norm) with a small number of samples. We coin the term `outlaw distributions' for such distributions since they `defy' the Law of Large Numbers. We show that the existence of outlaw distributions over sufficiently `smooth' functions implies the existence of constant query LDCs and vice versa. We give several candidates for outlaw distributions over smooth functions coming from finite field incidence geometry, additive combinatorics and from hypergraph (non)expanders. We also prove a useful lemma showing that (smooth) LDCs which are only required to work on average over a random message and a random message index can be turned into true LDCs at the cost of only constant factors in the parameters.Comment: A preliminary version of this paper appeared in the proceedings of ITCS 201

Affine extractors over large fields with exponential error

We describe a construction of explicit affine extractors over large finite fields with exponentially small error and linear output length. Our construction relies on a deep theorem of Deligne giving tight estimates for exponential sums over smooth varieties in high dimensions.Comment: To appear in Comput. Comple

Variety Evasive Sets

We give an explicit construction of a large subset of F^n, where F is a finite field, that has small intersection with any affine variety of fixed dimension and bounded degree. Our construction generalizes a recent result of Dvir and Lovett (STOC 2012) who considered varieties of degree one (affine subspaces).Comment: 13 page

A Sauer-Shelah-Perles Lemma for Sumsets

We show that any family of subsets $A\subseteq 2^{[n]}$ satisfies $\lvert A\rvert \leq O\bigl(n^{\lceil{d}/{2}\rceil}\bigr)$, where $d$ is the VC dimension of $\{S\triangle T \,\vert\, S,T\in A\}$, and $\triangle$ is the symmetric difference operator. We also observe that replacing $\triangle$ by either $\cup$ or $\cap$ fails to satisfy an analogous statement. Our proof is based on the polynomial method; specifically, on an argument due to [Croot, Lev, Pach '17].Comment: 6 pages, fixed a few typo

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

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 $S \subset \mathbb{F}_q^n$ and a randomly chosen linear map $L : \mathbb{F}_q^n \to \mathbb{F}_q^t$ with $q^t$ taken to be sufficiently smaller than $|S|$. Let $U_S$ denote a random variable distributed uniformly on $S$. Our main theorem shows that, with high probability over the choice of $L$, the random variable $L(U_S)$ is close to uniform in the $\ell_\infty$ norm. In other words, every element in the range $\mathbb{F}_q^t$ has about the same number of elements in $S$ mapped to it. This complements the widely-used Leftover Hash Lemma (LHL) which proves the analog statement under the statistical, or $\ell_1$, 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