Dragon curves in Littlewood roots

Abstract

A Littlewood polynomial is a polynomial whose coefficients lie in $\{- 1, +1\}$. While the majority of roots of a Littlewood polynomial of large degree are near the unit circle, numerical experiments suggest that when plotting the roots of \emph{all} Littlewood polynomials of a given large degree, striking fractal structures appear away from the unit circle. These fractals resemble the attractor of a certain iterated function system and are known as \emph{dragon curves}. In this note, we provide a rigorous explanation of this phenomenon, along with an analysis of a random variant, saying that such fractal behavior is typical.