Critical values for intermediate and box dimensions of projections and other images of compact sets

Abstract

Given a compact set $E\subset {\mathbb{R}}^d$, we investigate for which values of $m$ the equality ${\dim }_{\theta }{P}_V(E)=m$ or ${\dim }_{\theta }{P}_V(E)={\dim }_{\theta }E$ holds for ${\gamma }_{d,m}$-almost all $V\in G(d,m)$. Our result extends to more general functions, including orthogonal projections and fractional Brownian motion. As a particular case, when $\theta =1$, the results apply to the box dimension.