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 \(α>0\), there is a sequence of dual-lattice frequencies approaching the facet normal along which the Fourier coefficients of \(ρ_Λ^α\) are positive, with an explicit leading asymptotic determined by the facet. As direct consequences, Haar measure is not a local maximizer for any positive distance power, even among smooth densities, and every flat torus of dimension at least two has supremal negative type and generalized roundness zero. We then solve the global maximization problem for two classes of flat tori. On an orthogonal rectangular torus, the maximizers undergo a transition at \(α=2\): for \(1\leqα<2\) they are the translated uniform measures on the two-torsion subgroup; at \(α=2\) all balanced couplings on translates of that subgroup are extremal; and for \(α>2\) only equally weighted diametral pairs remain. On the regular hexagonal torus, the maximizers are precisely the uniform measures on translates of a distinguished cyclic subgroup of order three for every \(α\geq1\).