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

Given a compact set E \subset \mathbb{R}^{d} , we investigate for which values of m the equality \operatorname{dim}_{\theta} P_{V}(E) = m or \operatorname{dim}_{\theta} P_{V}(E) = \operatorname{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.