Typical Kakeya-type sets and their typical code sets

Abstract

We study properties of compact Besicovitch sets and other Kakeya-type sets which are typical in the sense that the exceptions form a meager set in the Baire category sense in a suitable metric space. The main motivation comes from the known fact that typical compact Besicovitch sets in $\mathbb{R}^n$ have minimal Hausdorff dimension among all Besicovitch sets of $\mathbb{R}^n$. Thus, in order to estimate the smallest possible Hausdorff dimension of an arbitrary Besicovitch set, it is enough to estimate the Hausdorff dimension of a typical (compact) Besicovitch set, or analogously of a Besicovitch set satisfying our list of typical properties. Among other results, we prove that a typical Besicovitch set $K$ in $\mathbb{R}^n$ has the following properties: it is not sticky, any line intersects $K$ in a set of ($1$-dimensional) measure zero or in a unit line segment and a set of measure zero, it has no more than $n$ different unit line segments in any direction, and for each $1\le j\le n$ the Hausdorff dimension of the set of those directions in which $K$ has exactly $j$ unit line segments is $n-j$. For compact metric spaces $X$ and $Y$, let $\mathcal{C}(X,Y)$ denote the set of compact subsets of $X\times Y$ whose projection onto the first coordinate is $X$, equipped with the Hausdorff metric. Kakeya-type compact sets can be coded by elements of such spaces, allowing typical properties to be studied through comeager subsets of $\mathcal C(X,Y)$. Among other results, we prove that, under suitable conditions on $X$ and $Y$, a typical $A\in \mathcal{C}(X,Y)$ is totally disconnected and purely unrectifiable, its projection to the second coordinate has Hausdorff dimension zero, and the dimension identities $\dim_H(A)=\dim_H(X)$ and $\dim_P(A)=\dim_P(X)+\dim Y$ hold.