Intermediate dimensions of slices of compact sets

TL;DR

This paper explores the relationship between the intermediate dimensions of fractal sets and their slices, establishing bounds, continuity properties, and introducing new measures to deepen understanding of fractal geometry.

Contribution

It introduces new bounds and continuity results for intermediate dimensions of slices, and develops a novel type of Frostman measure for these dimensions.

Findings

Upper bounds for intermediate dimensions of slices established.

Continuity of intermediate dimensions at  proven.

New Frostman measures for intermediate dimensions introduced.

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.