Kakeya books and projections of Kakeya sets

TL;DR

This paper investigates Kakeya sets, introducing Kakeya books with restricted line segment positions, and explores the connection between projection properties and the Kakeya conjecture, establishing equivalence conditions.

Contribution

It proves Kakeya books have full box dimension and links the projection behavior of Kakeya sets to the validity of the Kakeya conjecture.

Findings

Kakeya books have full box dimension.

Projection properties of Kakeya sets relate to the Kakeya conjecture.

Equivalence between projection independence and the conjecture.

Abstract

Here we show some results related with Kakeya conjecture which says that for any integer $n\geq 2$, a set containing line segments in every dimension in $\mathbb{R}^n$ has full Hausdorff dimension as well as box dimension. We proved here that the Kakeya books, which are Kakeya sets with some restrictions on positions of line segments have full box dimension. We also prove here a relation between the projection property of Kakeya sets and the Kakeya conjecture. If for any Kakeya set $K\subset\mathbb{R}^n$, the Hausdorff dimension of orthogonal projections on $k\leq n$ subspaces is independent of directions then the Kakeya conjecture is true. Moreover, the converse is also true.