Discrete analogues of Kakeya problems

Abstract

This thesis investigates two problems that are discrete analogues of two harmonic analytic problems which lie in the heart of research in the field. More specifically, we consider discrete analogues of the maximal Kakeya operator conjecture and of the recently solved endpoint multilinear Kakeya problem, by effectively shrinking the tubes involved in these problems to lines, thus giving rise to the problems of counting joints and multijoints with multiplicities. In fact, we effectively show that, in $\mathbb{R}^3$, what we expect to hold due to the maximal Kakeya operator conjecture, as well as what we know in the continuous case due to the endpoint multilinear Kakeya theorem by Guth, still hold in the discrete case. In particular, let $\mathfrak{L}$ be a collection of $L$ lines in $\mathbb{R}^3$ and $J$ the set of joints formed by $\mathfrak{L}$, that is, the set of points each of which lies in at least three non-coplanar lines of $\mathfrak{L}$. It is known that $|J|=O(L^{3/2})$ (first proved by Guth and Katz). For each joint $x\in J$, let the multiplicity $N(x)$ of $x$ be the number of triples of non-coplanar lines through $x$. We prove here that $$\sum_{x\in J} N(x)^{1/2}=O(L^{3/2}), $$while we also extend this result to real algebraic curves in $\mathbb{R}^3$ of uniformly bounded degree, as well as to curves in $\mathbb{R}^3$ parametrized by real univariate polynomials of uniformly bounded degree. The multijoints problem is a variant of the joints problem, involving three finite collections of lines in $\mathbb{R}^3$; a multijoint formed by them is a point that lies in (at least) three non-coplanar lines, one from each collection. We finally present some results regarding the joints problem in different field settings and higher dimensions.