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 $|\varphi_\lambda|^2$ 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 $S^n(\lambda)$ of radius $\sqrt{\lambda}$, and we use it in the proof.

Let $M$ be a closed Riemannian manifold. We consider the inner radius of a nodal domain for a large eigenvalue $\lambda$. We give upper and lower bounds on the inner radius of the type $C/\lambda^\alpha(\log\lambda)^\beta$. Our proof is based on a local behavior of eigenfunctions discovered by Donnelly and Fefferman and a Poincar\'{e} type inequality proved by Maz'ya. Sharp lower bounds are known only in dimension two. We give an account of this case too.