Polynomials with Zeros on the Unit Circle: Regularity of Leja Sequences

Let $z_1,\dots ,z_m$ be $m$ distinct complex numbers, normalized to $|z_k| = 1$,andconsiderthepolynomial$$p_m(z)={\prod }_{k=1}^m(z-z_k)\mathrm{.}$$ We define a sequence of polynomials in a greedy fashion, $$p_{N+1}(z)=p_N(z)(z-z^{\ast })where{z}^{\ast }=\arg \underset{\mathrm{\mid }z\mathrm{\mid }=1}{\max }\mathrm{\mid }{p}_N(z)\mathrm{\mid },$$ and prove that, independently of the initial polynomial $p_m$, the roots of $p_N$ equidistribute in angle at rate at most $(\log N{)}^2\mathrm{/}N$. This even persists when sometimes adding `adversarial’ points by hand. We rephrase the main result in terms of a dynamical system involving the inverse fractional Laplacian $(-\mathrm{\Delta }{)}^{-1\mathrm{/}2}$ and conjecture that, when phrased in this language, the underlying regularity phenomenon might appear in a very general setting.