  The Schur index of a $4$ dimensional $\mathcal{N}=2$ superconformal field
theory counts (with sign) bosonic and fermionic states that preserve $4$
supercharges. We consider the Schur indices of $4$d $\mathcal{N}=4$ super
Yang-Mills and $\mathcal{N}=2$ circular quiver gauge theories with gauge groups
$U(N)$ or $SU(N)$. We calculate the exponentially dominant part of their
asymptotic expansions as the index parameter $q$ approaches any root of unity.
We find that some of the indices exhibit ``small" ($\mathcal{O}(N^0)$ as $N
\rightarrow \infty$) exponential growth, which is much smaller than an
$\mathcal{O}(N^2)$ exponential growth of states that is indicative of a black
hole. This implies that the indices do not capture a growth of states that
would correspond to a supersymmetric black hole that preserves 4 supercharges
in the holographic dual AdS theory. Interestingly, the exponentially dominant
part in the Schur asymptotics we consider, depends on the parity of the rank
$N$.
