## Polynomials with coefficients from a finite set

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- by Peter Borwein, Tamás Erdélyi and Friedrich Littmann PDF
- Trans. Amer. Math. Soc.
**360**(2008), 5145-5154

## Abstract:

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 \begin{align*} K_n&:=\Big \{\sum _{k=0}^n a_k z^k : |a_0|=|a_n|=1\text { and }|a_k|\le 1\Big \} \end{align*} has at least \[ 8\sqrt {n}\log n \] zeros in any disk with center on the unit circle and radius $\delta _n$.

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## Additional Information

**Peter Borwein**- Affiliation: Department of Mathematics and Statistics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6
- Email: pborwein@cecm.sfu.ca
**Tamás Erdélyi**- Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
- Email: terdelyi@math.tamu.edu
**Friedrich Littmann**- Affiliation: Department of Mathematics, North Dakota State University, Fargo, North Dakota 58105
- Email: Friedrich.Littmann@ndsu.edu
- Received by editor(s): June 8, 2005
- Received by editor(s) in revised form: February 15, 2006
- Published electronically: May 27, 2008
- © Copyright 2008 by the authors
- Journal: Trans. Amer. Math. Soc.
**360**(2008), 5145-5154 - MSC (2000): Primary 30B30; Secondary 11C08, 30C15
- DOI: https://doi.org/10.1090/S0002-9947-08-04605-9
- MathSciNet review: 2415068