This paper studies the spectrum that results when all height one polynomials are evaluated at a Pisot number. This continues the research theme initiated by Erdős, Joó and Komornik in 1990. Of particular interest is the minimal non-zero value of this spectrum. Formally, this value is denoted as
$l^1(q)$
, and this definition is extended to all height
$m$
polynomials as
\[
l^m(q):= \inf(\vert y\vert: y = \epsilon_0 + \epsilon_1 q^1 + \cdots + \epsilon_n q^n,\,
\epsilon_i \in {\bb Z},\, \vert\epsilon_i\vert \leqslant m,\, y \ne 0).
\]
A recent result in 2000, of Komornik, Loreti and Pedicini gives a complete description of
$l^m(q)$
when
$q$
is the Golden ratio. This paper extends this result to include all unit quadratic Pisot numbers. A main theorem is as follows.
THEOREM. Let
$q$
be a quadratic Pisot number that satisfies a polynomial of the form
$p(x) = x^2 - ax \pm 1$
, with conjugate
$r$
. Let
$q$
have convergents
$\{C_k/D_k\}$
and let
$k$
be the maximal integer such that
\[
\vert D_k r - C_k\vert \leqslant m \frac{1}{1-\vert r\vert};
\]
then
\[
l^m (q) = \vert D_k q - C_k \vert.
\]
A value related to
$l(q)$
is
$a(q)$
, the minimal non-zero value when all
${\pm}1$
polynomials are evaluated at
$q$
. Formally, this is
\[
a(q) := \inf(\vert y\vert: y = \epsilon_0 + \epsilon_1 q^2 + \cdots + \epsilon_n q^n,\,
\epsilon_i = {\pm} 1,\, y \ne 0).
\]
An open question concerning how often
$a(q) = l(q)$
is also answered in this paper.