  We improve the recently discovered upper and lower bounds on the $O(1)$
correction to the Cardy formula for the density of states integrated over an
energy window (of width $2\delta$), centered at high energy in 2 dimensional
conformal field theory. We prove optimality of the lower bound for $\delta\to
1^{-}$. We prove a conjectured upper bound on the asymptotic gap between two
consecutive Virasoro primaries for a central charge greater than $1,$
demonstrating it to be $1.$ Furthermore, a systematic method is provided to
establish a limit on how tight the bound on the $O(1)$ correction to the Cardy
formula can be made using bandlimited functions. The techniques and the
functions used here are of generic importance whenever the Tauberian theorems
are used to estimate some physical quantities.
