Given a compact set E ⊂ Rᵈ , we investigate for which values of m the equality dimθ PV(E) = m or dimθ PV(E) = dimθ E holds for γd,m -almost all V ∈ G(d,m) . Our result extends to more general functions, including orthogonal projections and fractional Brownian motion. As a particular case, when θ = 1 , the results apply to the box dimension.
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Angelini et al. (2025) studied this question.