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We provide a proof of a conjecture by Jakobson, Nadirashvili, and Toth stating that on an 
$n$-dimensional flat torus 
${{\mathbb{T}}^{n}}$, and the Fourier transform of squares of the eigenfunctions 
${{\left| \varphi \lambda \right|}^{2}}$ of the Laplacian have uniform 
${{l}^{n}}$ bounds that do not depend on the eigenvalue 
$\lambda $. The proof is a generalization of an argument by Jakobson, et al. for the lower dimensional cases. These results imply uniform bounds for semiclassical limits on 
${{\mathbb{T}}^{n+2}}$. We also prove a geometric lemma that bounds the number of codimension-one simplices satisfying a certain restriction on an 
$n$-dimensional sphere 
${{S}^{n}}\,\left( \text{ }\!\!\lambda\!\!\text{ } \right)$ of radius 
$\sqrt{\lambda }$, and we use it in the proof.
