Marstrand-type projection theorems for the Assouad spectrum and quasi-Assouad dimension

Determine whether, for every Borel set E ⊂ ℝ^n, every 1 ≤ m < n, and every 0 < θ < 1, the orthogonal projection proj_V E onto an m-dimensional subspace V satisfies a Marstrand-type almost-sure formula for the Assouad spectrum; specifically, ascertain whether dim_A^θ(proj_V E) = min{m, dim_A^θ(E)} holds for γ_{n,m}-almost all V ∈ G(n,m), and similarly whether an almost-sure projection theorem holds for the quasi-Assouad dimension dim_qA(E).

Background

Assouad-type dimensions quantify the most extreme local scaling of sets. The Assouad spectrum dim_A^θ(E) (0 < θ < 1) interpolates between local coverings at different scales, and the quasi-Assouad dimension dim_qA(E) captures limiting behavior of the spectrum as θ → 1. For Hausdorff, box-counting, and packing dimensions there are Marstrand-type theorems stating that, for almost all orthogonal projections onto m-dimensional subspaces, the projected dimension equals the minimum of m and the original dimension (possibly via suitable dimension profiles).

In contrast, for the classical Assouad dimension there is no almost-sure Marstrand-type theorem, although lower bounds under projections are known. The survey explicitly notes that it is unknown whether analogous almost-sure projection results hold for the Assouad spectrum and the quasi-Assouad dimension, motivating a precise determination of such theorems.