Rigid-motion packing conjecture via spherical Fourier dimension threshold

Prove that for any Borel sets E ⊂ R^d and Θ ⊂ E(d) = O(d) × R^d, if the spherical Fourier dimension of E and the Hausdorff dimension of Θ satisfy dim_SF(E) + dim_H(Θ) > (d^2 + d)/2, then the Lebesgue measure of the union of rigid-motion images Θ(E) = {g(x) + z : (g, z) ∈ Θ, x ∈ E} is strictly positive, i.e., L^d(Θ(E)) > 0.

Background

This conjecture proposes a sharp positivity threshold for the Lebesgue measure of unions of rigid-motion copies of an arbitrary Borel set E in Euclidean space. The threshold is expressed in terms of the spherical Fourier dimension of E and the Hausdorff dimension of the set of rigid motions Θ ⊂ O(d) × R^d.

It is motivated by Theorem 1.7 and its corollary, which provide sufficient conditions involving product-structured motion sets and spherical Fourier dimension estimates, and by sharpness examples showing the necessity of such thresholds. The conjecture aims to extend these results to arbitrary Θ without product structure.