Semiclassical limits of eigenfunctions on flat $n$-dimensional tori

We provide a proof of the conjecture formulated in \cite{Jak97,JNT01} which states that on a $n$-dimensional flat torus ${\text{T}}^n$, the Fourier transform of squares of the eigenfunctions $\mathrm{\mid }{\varphi }_{\lambda }{\mathrm{\mid }}^2$ of the Laplacian have uniform $l^n$ bounds that do not depend on the eigenvalue $\lambda$. The proof is a generalization of the argument by Jakobson, {\it et al}. for the lower dimensional cases. These results imply uniform bounds for semiclassical limits on ${\text{TT}}^{n+2}$. We also prove a geometric lemma that bounds the number of codimension-one simplices which satisfy a certain restriction on an $n$-dimensional sphere $S^n(\lambda )$ of radius $\sqrt{\lambda }$ and use it in the proof.