Open Collections

Abstract

Kakeya sets are compact subsets of ℝⁿ that contain a unit line segment pointing in every direction. The Kakeya conjecture states that such sets must have Hausdorff dimension n. This thesis will be focused on studying the Kakeya conjecture. We shall firstly discuss the background associated to the Kakeya conjecture. In particular we shall give a brief survey of historical approaches to the problem. Then we proceed to prove my main result, which states that sticky Kakeya sets in ℝ⁴ have dimension at least 3.25. The property of stickiness was first discovered by Katz-Łaba-Tao in their 1999 breakthrough paper on the Kakeya problem. Then Wang-Zahl formalized the definition of a sticky Kakeya set, and they eventually went on to prove that the Kakeya conjecture is true for n = 3 in a spectacular sequence of recent papers. For n = 4, the best current general bound (that is, without stickiness assumption) is due to Katz and Zahl. In particular, Katz-Zahl used the planebrush argument to show that Kakeya sets in ℝ⁴ have Hausdorff dimension at least 3.059. A planebrush is a geometric object that is a higher dimensional analogue of a geometric object called a hairbrush, which was introduced by Wolff. We wish to apply the planebrush argument to sticky Kakeya sets, as these are an important sub- class to study. Moreover, when we restrict our attention to sticky Kakeya sets, we can improve upon the Katz-Zahl bound by combining the plane-brush result with multi-scale structural information coming from stickiness.