New Kakeya estimates using Gromov's algebraic lemma

This paper presents several new results related to the Kakeya problem. First, we establish a geometric inequality which says that collections of direction-separated tubes (thin neighborhoods of line segments that point in different directions) cannot cluster inside thin neighborhoods of low degree algebraic varieties. We use this geometric inequality to obtain a new family of multilinear Kakeya estimates for direction-separated tubes. Using the linear / multilinear theory of Bourgain and Guth, these multilinear Kakeya estimates are converted into Kakeya maximal function estimates. Specifically, we obtain a Kakeya maximal function estimate in $\mathbb{R}^n$ at dimension $d(n) = (2-\sqrt{2})n + c(n)$ for some $c(n)>0$. Our bounds are new in all dimensions except $n=2,3,4,$ and $6$.