Let $a_n\in \mathbb{R}$ be such that ${\sum }_n|a_n{|}^2=\infty$ and $|a_n|=o(1/\sqrt{n})$. Is it true that, for almost all ${ϵ}_n=\pm 1$, there exists some $z$ with $|z|=1$ (depending on the choice of signs) such that$$\sum _n{ϵ}_n{a}_n{z}^n$$converges?