Polynomials with coefficients from a finite set

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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper C"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {C}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper C"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {C}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be so large that <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="delta Subscript n Baseline colon equals 33 pi StartFraction log n Over StartRoot n EndRoot EndFraction"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi> δ </mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>:=</mml:mo> <mml:mn>33</mml:mn> <mml:mi> π </mml:mi> <mml:mfrac> <mml:mrow> <mml:mi>log</mml:mi> <mml:mo> ⁡ </mml:mo> <mml:mi>n</mml:mi> </mml:mrow> <mml:msqrt> <mml:mi>n</mml:mi> </mml:msqrt> </mml:mfrac> </mml:mrow> <mml:annotation encoding="application/x-tex">\delta _n:=33\pi \frac {\log n}{\sqrt {n}}</mml:annotation> </mml:semantics> </mml:math> \] </disp-formula> satisfies <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="delta Subscript n Baseline less-than-or-equal-to 1"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi> δ </mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo> ≤ </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\delta _n\le 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We show that any polynomial in <disp-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row 1st Column upper K Subscript n 2nd Column a m p semicolon colon equals StartSet sigma-summation Underscript k equals 0 Overscript n Endscripts a Subscript k Baseline z Superscript k Baseline colon StartAbsoluteValue a 0 EndAbsoluteValue equals StartAbsoluteValue a Subscript n Baseline EndAbsoluteValue equals 1 and StartAbsoluteValue a Subscript k Baseline EndAbsoluteValue less-than-or-equal-to 1 EndSet EndLayout"> <mml:semantics> <mml:mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" side="left" displaystyle="true"> <mml:mtr> <mml:mtd> <mml:msub> <mml:mi>K</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mtd> <mml:mtd> <mml:mi>a</mml:mi> <mml:mi>m</mml:mi> <mml:mi>p</mml:mi> <mml:mo>;</mml:mo> <mml:mo>:=</mml:mo> <mml:mstyle scriptlevel="0"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo maxsize="1.623em" minsize="1.623em">{</mml:mo> </mml:mrow> </mml:mstyle> <mml:munderover> <mml:mo> ∑ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>k</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:mi>n</mml:mi> </mml:munderover> <mml:msub> <mml:mi>a</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:msup> <mml:mi>z</mml:mi> <mml:mi>k</mml:mi> </mml:msup> <mml:mspace width="thinmathspace"/> <mml:mo>:</mml:mo> <mml:mspace width="thinmathspace"/> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:msub> <mml:mi>a</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:msub> <mml:mi>a</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mtext> and </mml:mtext> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:msub> <mml:mi>a</mml:mi> <mml:mi>k</mml:mi> </mml:msub>