Distance Energies and Negative Type of Flat Tori

Let \(T_Λ=\mathbb R^d/Λ\), \(d\geq2\), be a flat torus with quotient metric \(ρ_Λ\), and consider the distance energies \(I_α(μ)=\iint ρ_Λ(x,y)^α\,dμ(x)\,dμ(y)\) of Borel probability measures \(μ\). We prove a quantitative Fourier signature of the cut locus: for every Voronoi facet and every \(α>0\)