We improve Frank-Sabin's work concerning the spectral cluster bounds for orthonormal systems at p=∞, on the flat torus and spaces of nonpositive sectional curvature, by shrinking the spectral band from [λ2,(λ+1)2)
Nearby in the stack
to
[λ2,(λ+ε(λ))2)
, where
ε(λ)
is a function of
λ
that goes to
0
as
λ
goes to
∞
. In achieving this, we invoke the method developed by Bourgain-Shao-Sogge-Yao.