Tag - Kakeya problem

Zeev Dvir: High-Dimensional Variants of the Finite Field Kakeya Problem

23rd October, 2023

The finite field Kakeya problem asks about the size of the smallest set in (𝔽q)n containing a line in every direction. Raised by Wolff in 1999 as a 'toy' version of the Euclidean Kakeya conjecture, this problem is now completely resolved using the polynomial method. In this talk I will describe recent progress on its higher-dimensional variant in which lines are replaced with k-dimensional flats. It turns out that, unlike in the 1-dimensional case, when k ≥ 2, one can prove that there are no 'interesting' constructions (with size smaller than trivial) even if one asks for sets that only have large intersection with a flat in every direction. This theorem turns out to have surprising applications in questions involving lattice coverings and linear hash functions.

Barak Weiss: New bounds on lattice covering volumes, and nearly uniform covers

25th May, 2023

Let L be a lattice in ℝn and let K be a convex body. The covering volume of L with respect to K is the minimal volume of a dilate rK, such that L+rK = ℝn, normalized by the covolume of L. Pairs (L,K) with small covering volume correspond to efficient coverings of space by translates of K, where the translates lie in a lattice. Finding upper bounds on the covering volume as the dimension n grows is a well studied problem in the so-called 'Geometry of Numbers', with connections to practical questions arising in computer science and electrical engineering. In a recent paper with Or Ordentlich (EE, Hebrew University) and Oded Regev (CS, NYU) we obtain substantial improvements to bounds of Rogers from the 1950s. In another recent paper, we obtain bounds on the minimal volume of nearly uniform covers (to be defined in the talk). The key to these results are recent breakthroughs by Dvir and others regarding the discrete Kakeya problem. I will give an overview of the questions and results.

Marina Iliopoulou: A discrete Kakeya-type inequality

22nd June, 2020

The Kakeya conjectures of harmonic analysis claim that congruent tubes that point in different directions rarely meet. In this talk we discuss the resolution of an analogous problem in a discrete setting (where the tubes are replaced by lines), and provide some structural information on quasi-extremal configurations.