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We improve Frank-Sabin's work concerning the spectral cluster bounds fororthonormal systems at $p=\infty$, on the flat torus and spaces of nonpositivesectional curvature, by shrinking the spectral band from $[\lambda^{2},(\lambda+1)^{2})$ to $[\lambda^{2}, (\lambda+\epsilon(\lambda))^{2})$, where$\epsilon(\lambda)$ is a function of $\lambda$ that goes to $0$ as $\lambda$goes to $\infty$. In achieving this, we invoke the method developed byBourgain-Shao-Sogge-Yao.

Improved Spectral Cluster Bounds for Orthonormal Systems