Zero-free neighborhood around the unit circle for Kac polynomials

In this paper, we study how the roots of the so-called Kac polynomial $W_n(z) = \sum_{k=0}^{n-1} \xi_k z^k$ are concentrating to the unit circle when its coefficients of $W_n$ are independent and identically distributed non-degenerate real random variables. It is well-known that the roots of a Kac polynomial are concentrating around the unit circle as $n\to \mathrm{\infty }$ if and only if $\mathbb{E}[\log (1+\mathrm{\mid }{\xi }_0\mathrm{\mid })]<\mathrm{\infty }$. Under the condition of $\mathbb{E}[{\xi }_0^2]<\mathrm{\infty }$, we show that there exists an annulus of width $\mathrm{O}(n^{-2}(\log n{)}^{-3})$ around the unit circle which is free of roots with probability $1-\mathrm{O}((\log n{)}^{-1\mathrm{/}2})$. The proof relies on the so-called small ball probability inequalities and the least common denominator.