Bounds on the dimension of lineal extensions

TL;DR

This paper proves a strong packing dimension version of Keleti's line segment extension conjecture in the plane, establishing links to the Kakeya conjecture and providing estimates in higher dimensions.

Contribution

It introduces a novel packing dimension approach to Keleti's conjecture and derives implications for the Kakeya problem in the plane and higher dimensions.

Findings

Proved a packing dimension variant of Keleti's conjecture in .

Established connections to the generalized Kakeya conjecture for packing dimension.

Provided doubling estimates and related results in higher dimensions.

Abstract

Let $E \subseteq \mathbb{R}^n$ be a union of line segments and $F \subseteq \mathbb{R}^n$ the set obtained from $E$ by extending each line segment in $E$ to a full line. Keleti's line segment extension conjecture posits that the Hausdorff dimension of $F$ should equal that of $E$. Working in $\mathbb{R}^2$, we use effective methods to prove a strong packing dimension variant of this conjecture, from which the generalized Kakeya conjecture for packing dimension immediately follows. This is followed by several doubling estimates in higher dimensions and connections to related problems.