Scaling limit for the kernel of the spectral projector and remainder estimates in the pointwise weyl law

Let (M; g) be a compact, smooth, Riemannian manifold. We obtain new off-diagonal estimates as λ→1 for the remainder in the pointwise Weyl law for the kernel of the spectral projector of the Laplacian onto functions with frequency at most λ. A corollary is that, when rescaled around a non-self-focal point, the kernel of the spectral projector onto the frequency interval (λ, λ+1] has a universal scaling limit as λ→(depending only on the dimension of M). Our results also imply that, if M has no conjugate points, then immersions of M into Euclidean space by an orthonormal basis of eigenfunctions with frequencies in.(λ, λ+1] are embeddings for all λ sufficiently large.