1. CJM Online first
 Hare, Kathryn; Hare, Kevin; Ng, Michael Ka Shing

Local dimensions of measures of finite type II  Measures without full support and with nonregular probabilities
Consider a finite sequence of linear contractions $S_{j}(x)=\varrho
x+d_{j}$ and
probabilities $p_{j}\gt 0$ with $\sum p_{j}=1$. We are interested
in the
selfsimilar measure $\mu =\sum p_{j}\mu \circ S_{j}^{1}$, of
finite type.
In this paper we study the multifractal analysis of such measures,
extending the theory to measures arising from nonregular probabilities
and
whose support is not necessarily an interval.
Under some mild technical assumptions, we prove that there exists
a subset
of supp$\mu $ of full $\mu $ and Hausdorff measure, called the
truly
essential class, for which the set of (upper or lower) local
dimensions is a
closed interval. Within the truly essential class we show that
there exists
a point with local dimension exactly equal to the dimension of
the support.
We give an example where the set of local dimensions is a two
element set,
with all the elements of the truly essential class giving the
same local
dimension. We give general criteria for these measures to be
absolutely
continuous with respect to the associated Hausdorff measure of
their support
and we show that the dimension of the support can be computed
using only
information about the essential class.
To conclude, we present a detailed study of three examples. First,
we show
that the set of local dimensions of the biased Bernoulli convolution
with
contraction ratio the inverse of a simple Pisot number always
admits an
isolated point. We give a precise description of the essential
class of a
generalized Cantor set of finite type, and show that the $kth$
convolution
of the associated Cantor measure has local dimension at $x\in
(0,1)$ tending
to 1 as $k$ tends to infinity. Lastly, we show that within a
maximal loop
class that is not truly essential, the set of upper local dimensions
need
not be an interval. This is in contrast to the case for finite
type measures
with regular probabilities and full interval support.
Keywords:multifractal analysis, local dimension, IFS, finite type Categories:28A80, 28A78, 11R06 

2. CJM 2014 (vol 67 pp. 507)
 Borwein, Peter; Choi, Stephen; Ferguson, Ron; Jankauskas, Jonas

On Littlewood Polynomials with Prescribed Number of Zeros Inside the Unit Disk
We investigate the numbers of complex zeros of Littlewood polynomials
$p(z)$ (polynomials with coefficients $\{1, 1\}$) inside or
on the unit circle $z=1$, denoted by $N(p)$ and $U(p)$, respectively.
Two types of Littlewood polynomials are considered: Littlewood
polynomials with one sign change in the sequence of coefficients
and Littlewood polynomials with one negative coefficient. We
obtain explicit formulas for $N(p)$, $U(p)$ for polynomials $p(z)$
of these types. We show that, if $n+1$ is a prime number, then
for each integer $k$, $0 \leq k \leq n1$, there exists a Littlewood
polynomial $p(z)$ of degree $n$ with $N(p)=k$ and $U(p)=0$. Furthermore,
we describe some cases when the ratios $N(p)/n$ and $U(p)/n$
have limits as $n \to \infty$ and find the corresponding limit
values.
Keywords:Littlewood polynomials, zeros, complex roots Categories:11R06, 11R09, 11C08 

3. CJM 2012 (vol 64 pp. 254)
4. 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 

5. CJM 2010 (vol 62 pp. 543)
 Hare, Kevin G.

More Variations on the SierpiÅski Sieve
This paper answers a question of Broomhead, Montaldi and Sidorov about the existence of gaskets of a particular type related to the SierpiÅski sieve. These gaskets are given by iterated function systems that do not satisfy the open set condition. We use the methods of Ngai and Wang to compute the dimension of these gaskets.
Categories:28A80, 28A78, 11R06 

6. 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 

7. CJM 2002 (vol 54 pp. 468)
 Boyd, David W.; RodriguezVillegas, Fernando

Mahler's Measure and the Dilogarithm (I)
An explicit formula is derived for the logarithmic Mahler measure
$m(P)$ of $P(x,y) = p(x)y  q(x)$, where $p(x)$ and $q(x)$ are
cyclotomic. This is used to find many examples of such polynomials
for which $m(P)$ is rationally related to the Dedekind zeta value
$\zeta_F (2)$ for certain quadratic and quartic fields.
Categories:11G40, 11R06, 11Y35 
