Critical values for Intermediate and Box dimension of projections and other images
Nicolas Angelini, Ursula Molter
TL;DR
This work develops a Marstrand‑type theory for how θ‑intermediate and box dimensions behave under orthogonal projections of a compact set E ⊂ ℝ^d. By introducing the lower and upper quasi‑Hausdorff dimensions dim_{qH} E and dimension profiles, the authors prove that for almost every projection, the quasi‑Hausdorff dimensions of the image satisfy $\underline{\dim}_{qH} P_V E = \min\{m, \underline{\dim}_{qH} E\}$ and $\overline{\dim}_{qH} P_V E = \min\{m, \overline{\dim}_{qH} E\}$, with direct consequences for the box dimension. The results extend to index‑α fractional Brownian motion images, establishing precise formulae $\underline{\dim}_{\theta} B_α(E) = (1/α)\underline{\dim}_{\theta}^{mα} E$ and $\overline{\dim}_{\theta} B_α(E) = (1/α)\overline{\dim}_{\theta}^{mα} E$, and tie the behavior to Assouad spectra to derive lower bounds and preservation results under projections. Collectively, the paper provides a broad, quantitative framework for how projection and image operations affect a wide spectrum of fractal dimensions, with implications for geometric measure theory and stochastic image models such as fractional Brownian motion.
Abstract
Given a compact set $E\subset\mathbb{R}^d$ we investigate for which values of $m$ we have that $\dim_θP_V(E)=m$ or $\dim_θP_V(E)=\dim_θE$ for $γ_{d,m}-$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 $θ=1$, the results are valid for the Box dimension.