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

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.