Marstrand-type projection theorem for Fourier dimension

Establish whether, for every Borel set E ⊂ ℝ^n and every 1 ≤ m < n, the Fourier dimension of the orthogonal projection proj_V E equals min{m, dim_F(E)} for γ_{n,m}-almost all V ∈ G(n,m); that is, determine if a Marstrand-type almost-sure projection result holds for Fourier dimension.

Background

Fourier dimension captures decay of the Fourier transform of measures supported on a set, and it always satisfies dim_F(E) ≤ dim_H(E). Under projection, the Fourier transform interacts naturally with restriction to subspaces, yielding general inequalities min{m, dim_F(E)} ≤ dim_F(proj_V E) ≤ min{m, dim_H(E)} for all V. For Salem sets (where dim_F(E) = dim_H(E)), equality holds for all projection directions.

Despite these inequalities and special cases, the survey emphasizes that it is unknown whether a Marstrand-type almost-sure projection theorem holds in general for Fourier dimension. Resolving this would align the Fourier dimension theory with the well-established almost-sure projection results for Hausdorff, packing, and box-counting dimensions.