Salem–Zygmund inequality for locally sub-Gaussian random variables, random trigonometric polynomials, and random circulant matrices

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