The sequence of partial sums of a unimodular power series is not ultraflat

Abstract

We show that if $(a_j{)}_{j=0}^{\infty }$ is a sequence of numbers $a_j\in \mathbb{C}$ with $|a_j|=1$, and $$P_n(z)=\sum _{j=0}^n{a}_j{z}^j,n=0,1,2,\dots ,$$ then $(P_n)$ is NOT an ultraflat sequence of unimodular polynomials. This answers a question raised by Zachary Chase.