  We improve Frank-Sabin's work concerning the spectral cluster bounds for
orthonormal systems at $p=\infty$, on the flat torus and spaces of nonpositive
sectional 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 by
Bourgain-Shao-Sogge-Yao.
