Critical values for Intermediate and Box dimension of projections and other images

TL;DR

This paper studies the behavior of intermediate and Box dimensions of projections of compact sets in Euclidean space, identifying conditions under which these dimensions attain specific values for almost all projections.

Contribution

It extends existing results to more general functions, including orthogonal projections and fractional Brownian motion, providing new insights into dimension behavior.

Findings

Dimension of projections equals m for almost all V in G(d,m).

Results apply to Box dimension as a special case.

Extension to fractional Brownian motion functions.

Abstract

Given a compact set $E\subset\mathbb{R}^d$ we investigate for which values of $m$ we have that $\dim_\theta P_V(E)=m$ or $\dim_\theta P_V(E)=\dim_\theta E$ for $\gamma_{d,m}-$almost all $V\in G(d,m)$. Our result can be extended to more general functions that include orthogonal projections and fractional Brownian motion. As a particular case, letting $\theta=1$, the results are valid for the Box dimension.