Georgia Institute of Technology College of Sciences

Kakeya sets are compact subsets of $\mathbb{R}^n$ that contain a unit line segment pointing in every direction and the Kakeya conjecture states that such sets must have Hausdorff dimension $n$. The property of stickiness was first discovered by by Katz-Laba-Tao in their 1999 breakthrough paper on the Kakeya problem. Then Wang-Zahl formalized the definition of a sticky Kakeya set as a subclass of general Kakeya sets in 2022. Sticky Kakeya sets played an important role as Wang and Zahl solved the Kakeya conjecture for  $\mathbb{R}^3$ in a major recent development.
The planebrush method is a geometric argument by Katz-Zahl which gives the current best bound of 3.059 for Hausdorff dimension of Kakeya sets in $\mathbb{R}^4$. Our new result shows that sticky Kakeya sets in $\mathbb{R}^4$ have dimension 3.25. The planebrush argument when combined with the sticky hypothesis gives us this better bound.