Intermediate dimensions of slices of compact sets

Abstract

$\theta$ intermediate dimensions are a continuous family of dimensions that interpolate between Hausdorff and Box dimensions of fractal sets. In this paper we study the problem of the relationship between the dimension of a set $E\subset\mathbb{R}^d$ and the dimension of the slices $E\cap V$ from the point of view of intermediate dimensions. Here $V\in G(d,m)$, where $G(d,m)$ is the set of $m-$dimensional subspaces of $\mathbb{R}^d$. We obtain upper bounds analogous to those already known for Hausdorff dimension. In addition, we prove several corollaries referring to, among other things, the continuity of these dimensions at $\theta=0$, a natural problem that arises when studying them. We also investigate which conditions are sufficient to obtain a lower bound that provides an equality for almost all slices. Finally, a new type of Frostman measures is introduced. These measures combine the results already known for intermediate dimensions and Frostman measures in the case of Hausdorff dimension.