Polynomials with coefficients from a finite set

In 1945 Duffin and Schaeffer proved that a power series that is bounded in a sector and has coefficients from a finite subset of $\mathbb{C}$ is already a rational function. Their proof is relatively indirect. It is one purpose of this paper to give a shorter direct proof of this beautiful and surprising theorem.

This will allow us to give an easy proof of a recent result of two of the authors stating that a sequence of polynomials with coefficients from a finite subset of $\mathbb{C}$ cannot tend to zero uniformly on an arc of the unit circle.

Another main result of this paper gives explicit estimates for the number and location of zeros of polynomials with bounded coefficients. Let $n$ be so large that

$$\delta_n:=33\pi \frac{\log n}{\sqrt{n}}$$

satisfies $\delta_n\le 1$. We show that any polynomial in

$K_n$

$:=\Big\{\sum_{k=0}^n a_k z^k\,:\,\vert a_0\vert=\vert a_n\vert=1$ and $\vert a_k\vert\le 1\Big\}$

has at least

$$8\sqrt{n}\log n$$

zeros in any disk with center on the unit circle and radius $\delta_n$.