Dimension of Heisenberg Kakeya sets and Circular Furstenberg sets
The Kakeya conjecture is one of the most famous open problems in geometric measure theory. It claims that a set in d-dimensional Euclidean space that contains a unit line segment in every direction necessarily has Hausdorff dimension d. It was resolved in the plane in 1970s, while remains open in higher dimensions. The conjecture has been readily found to be related to many problems in other fields, including harmonic analysis, incidence geometry and additive combinatorics. The study of these problems has become a central topic in related areas and is developing very fast recently.
Defining Kakeya sets in the first Heisenberg group
In his dissertation Jiayin Liu focuses on two variants of the Kakeya conjecture. For the first variant, Liu introduces the concept of a Heisenberg Kakeya set, and proves a sharp lower bound on the Heisenberg Hausdorff dimension of such a set in the first Heisenberg group.
— Compared to the Kakeya conjecture for Euclidean Kakeya sets, the core difference is that, instead of containing a line segment in every direction, it only requires to contain every horizontal line segment for a Heisenberg Kakeya set, explains Liu.
To show the desired dimension estimate, Liu introduces a parametrization of horizontal lines which converts the question to the one about restricted projections established by Käenmäki-Orponen-Venieri.
Defining circular Furstenberg sets
For the second variant, Liu introduces the notion of a circular Furstenberg set by replacing lines with circles in the classical Kakeya sets and obtains three dimension lower bounds for Furstenberg sets.
— The first estimate follows from a basic geometric fact (three non-collinear points determine a unique circle) together with a double-counting argument motivated by Wolff’s work on planar Kakeya sets. The second estimate is based on a deep result of circular Kakeya sets studied by Schlag and Wolff, describes Liu.
The third estimate is a joint work with his doctoral supervisors Katrin Fässler and Tuomas Orponen and this estimate is sharp. They deeply investigate Wolff’s upper bound involving the number of tangencies for a given family of circles and provide a useful generalization of this bound, which, ultimately, enables them to exploit an induction on the “degree of tangency of circles” and several iterative arguments to establish the desired result.
The examination of Jiayin Liu’s doctoral thesis “Dimension of Heisenberg Kakeya sets and Circular Furstenberg sets” will be held on 19.4.2024 at 12.00 in Mattilanniemi, room MaA211. Opponent is Associate Professor Ville Suomala (University of Oulu) and custos is Associate Professor Tuomas Orponen (Ģֱ). The doctoral dissertation is held in English.
Publishing information
The dissertation “Dimension of Heisenberg Kakeya sets and Circular Furstenberg sets” can be read on the JYX publication archive: