  In this manuscript we give an extension of the classic Salem--Zygmund
inequality for locally sub-Gaussian random variables. As an application, the
concentration of the roots of a Kac polynomial is studied, which is the main
contribution of this manuscript. More precisely, we assume the existence of the
moment generating function for the iid random coefficients for the Kac
polynomial and prove that there exists an annulus of width \[O(n^{-2}(\log
n)^{-1/2-\gamma}), \quad \gamma>1/2\] around the unit circle that does not
contain roots with high probability. As an another application, we show that
the smallest singular value of a random circulant matrix is at least
$n^{-\rho}$, $\rho\in(0,1/4)$ with probability $1-O(n^{-2\rho})$.
