Trace and Observability Inequalities for Laplace Eigenfunctions on the Torus

We investigate trace and observability inequalities for Laplace eigenfunctions on the d-dimensional torus, with respect to arbitrary Borel measures μ. Specifically, we characterize the measures μ for which the inequalities |u|² d μ |u|² d x (trace), |u|² d μ |u|² d x (observability) hold uniformly for all eigenfunctions u of the Laplacian. Sufficient conditions are derived based on the integrability and regularity of μ, while necessary conditions are formulated in terms of the dimension of the support of the measure. These results generalize classical theorems of Zygmund and Bourgain--Rudnick to higher dimensions. Applications include results in the spirit of Cantor--Lebesgue theorems, constraints on quantum limits, and control theory for the Schrödinger equation. Our approach combines several tools: the cluster structure of lattice points on spheres; decoupling estimates; and the construction of eigenfunctions exhibiting strong concentration or vanishing behavior, tailored respectively to the trace and observability inequalities.