Polynomials with coefficients from a finite set

Authors:
Peter Borwein, Tamás Erdélyi and Friedrich Littmann

Journal:
Trans. Amer. Math. Soc. **360** (2008), 5145-5154

MSC (2000):
Primary 30B30; Secondary 11C08, 30C15

Published electronically:
May 27, 2008

MathSciNet review:
2415068

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Abstract | References | Similar Articles | Additional Information

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 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 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 be so large that

and |

has at least

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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

DOI:
https://doi.org/10.1090/S0002-9947-08-04605-9

Keywords:
Zeros,
rational functions,
Duffin--Schaeffer Theorem,
Littlewood polynomials

Received by editor(s):
June 8, 2005

Received by editor(s) in revised form:
February 15, 2006

Published electronically:
May 27, 2008

Article copyright:
© Copyright 2008
by the authors