1. CJM 2012 (vol 64 pp. 254)
2. CJM 2011 (vol 64 pp. 345)
 McKee, James; Smyth, Chris

Salem Numbers and Pisot Numbers via Interlacing
We present a general construction of Salem numbers via rational
functions whose zeros and poles mostly lie on the unit circle and
satisfy an interlacing condition. This extends and unifies earlier
work. We then consider the ``obvious'' limit points of the set of Salem
numbers produced by our theorems and show that these are all Pisot
numbers, in support of a conjecture of Boyd. We then show that all
Pisot numbers arise in this way. Combining this with a theorem of
Boyd, we produce all Salem numbers via an interlacing construction.
Keywords:Salem numbers, Pisot numbers Category:11R06 

3. CJM 2009 (vol 61 pp. 264)
 Bell, J. P.; Hare, K. G.

On $\BbZ$Modules of Algebraic Integers
Let $q$ be an algebraic integer of degree $d \geq 2$.
Consider the rank of the multiplicative subgroup of $\BbC^*$ generated
by the conjugates of $q$.
We say $q$ is of {\em full rank} if either the rank is $d1$ and $q$
has norm $\pm 1$, or the rank is $d$.
In this paper we study some properties of $\BbZ[q]$ where $q$ is an
algebraic integer of full rank.
The special cases of when $q$ is a Pisot number and when $q$ is a Pisotcyclotomic number
are also studied.
There are four main results.
\begin{compactenum}[\rm(1)]
\item If $q$ is an algebraic integer of full rank and $n$ is a fixed positive
integer,
then there are only finitely many $m$ such that
$\disc\left(\BbZ[q^m]\right)=\disc\left(\BbZ[q^n]\right)$.
\item If $q$ and $r$ are algebraic integers of degree $d$ of full rank
and $\BbZ[q^n] = \BbZ[r^n]$ for
infinitely many $n$, then either $q = \omega r'$ or $q={\rm Norm}(r)^{2/d}\omega/r'$,
where
$r'$ is some conjugate of $r$ and $\omega$ is some root of unity.
\item Let $r$ be an algebraic integer of degree at most $3$.
Then there are at most $40$ Pisot numbers $q$ such that
$\BbZ[q] = \BbZ[r]$.
\item There are only finitely many Pisotcyclotomic numbers of any fixed
order.
\end{compactenum}
Keywords:algebraic integers, Pisot numbers, full rank, discriminant Categories:11R04, 11R06 

4. CJM 1997 (vol 49 pp. 887)
 Borwein, Peter; Pinner, Christopher

Polynomials with $\{ 0, +1, 1\}$ coefficients and a root close to a given point
For a fixed algebraic number $\alpha$ we
discuss how closely $\alpha$ can be approximated by
a root of a $\{0,+1,1\}$ polynomial of given degree.
We show that the worst rate of approximation tends to
occur for roots of unity, particularly those of small degree.
For roots of unity these bounds depend on
the order of vanishing, $k$, of the polynomial at $\alpha$.
In particular we obtain the following. Let
${\cal B}_{N}$ denote the set of roots of all
$\{0,+1,1\}$ polynomials of degree at most $N$ and
${\cal B}_{N}(\alpha,k)$ the roots of those
polynomials that have a root of order at most $k$
at $\alpha$. For a Pisot number $\alpha$ in $(1,2]$
we show that
\[
\min_{\beta \in {\cal B}_{N}\setminus \{ \alpha \}} \alpha
\beta \asymp \frac{1}{\alpha^{N}},
\]
and for a root of unity $\alpha$ that
\[
\min_{\beta \in {\cal B}_{N}(\alpha,k)\setminus \{\alpha\}}
\alpha \beta\asymp \frac{1}{N^{(k+1) \left\lceil
\frac{1}{2}\phi (d)\right\rceil +1}}.
\]
We study in detail the case of $\alpha=1$, where, by far, the
best approximations are real.
We give fairly precise bounds on the closest real root to 1.
When $k=0$ or 1 we
can describe the extremal polynomials explicitly.
Keywords:Mahler measure, zero one polynomials, Pisot numbers, root separation Categories:11J68, 30C10 
