Dimension estimates for Kakeya sets defined in an axiomatic setting

Abstract

In this dissertation we define a generalization of Kakeya sets in certain metric spaces. Kakeya sets in Euclidean spaces are sets of zero Lebesgue measure containing a segment of length one in every direction. A famous conjecture, known as Kakeya conjecture, states that the Hausdorff dimension of any Kakeya set should equal the dimension of the space. It was proved only in the plane, whereas in higher dimensions both geometric and arithmetic combinatorial methods were used to obtain partial results. In the first part of the thesis we define generalized Kakeya sets in metric spaces satisfying certain axioms. These allow us to prove some lower bounds for the Hausdorff dimension of generalized Kakeya sets using two methods introduced in the Euclidean context by Bourgain and Wolff. With this abstract setup we can deal with many special cases in a unified way, recovering some known results and proving new ones. In the second part we present various applications. We recover some of the known estimates for the classical Kakeya and Nikodym sets and for curved Kakeya sets. Moreover, we prove lower bounds for the dimension of sets containing a segment in a line through every point of a hyperplane and of an (n-1)-rectifiable set. We then show dimension estimates for Furstenberg type sets (already known in the plane) and for the classical Kakeya sets with respect to a metric that is homogeneous under non-isotropic dilations and in which balls are rectangular boxes with sides parallel to the coordinate axis. Finally, we prove lower bounds for the classical bounded Kakeya sets and a natural modification of them in Carnot groups of step two whose second layer has dimension one, such as the Heisenberg group. On the other hand, if the dimension is bigger than one we show that we cannot use this approach.