1. CJM 2014 (vol 67 pp. 795)
 Di Nasso, Mauro; Goldbring, Isaac; Jin, Renling; Leth, Steven; Lupini, Martino; Mahlburg, Karl

On a Sumset Conjecture of ErdÅs
ErdÅs conjectured that for any set $A\subseteq \mathbb{N}$
with positive
lower asymptotic density, there are infinite sets $B,C\subseteq
\mathbb{N}$
such that $B+C\subseteq A$. We verify ErdÅs' conjecture in
the case that $A$ has Banach density exceeding $\frac{1}{2}$.
As a consequence, we prove that, for $A\subseteq \mathbb{N}$
with
positive Banach density (a much weaker assumption than positive
lower density), we can find infinite $B,C\subseteq \mathbb{N}$
such
that $B+C$ is contained in the union of $A$ and a translate of
$A$. Both of the aforementioned
results are generalized to arbitrary countable
amenable groups. We also provide a positive solution to ErdÅs'
conjecture for subsets of the natural numbers that are pseudorandom.
Keywords:sumsets of integers, asymptotic density, amenable groups, nonstandard analysis Categories:11B05, 11B13, 11P70, 28D15, 37A45 

2. CJM 2012 (vol 64 pp. 254)
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 2008 (vol 60 pp. 1267)
 Blake, Ian F.; Murty, V. Kumar; Xu, Guangwu

Nonadjacent Radix$\tau$ Expansions of Integers in Euclidean Imaginary Quadratic Number Fields
In his seminal papers, Koblitz proposed curves
for cryptographic use. For fast operations on these curves,
these papers also
initiated a study of the radix$\tau$ expansion of integers in the number
fields $\Q(\sqrt{3})$ and $\Q(\sqrt{7})$. The (window)
nonadjacent form of $\tau$expansion of integers in
$\Q(\sqrt{7})$ was first investigated by Solinas.
For integers in $\Q(\sqrt{3})$, the nonadjacent form
and the window nonadjacent form of the $\tau$expansion were
studied. These are used for efficient
point multiplications on Koblitz curves.
In this paper, we complete
the picture by producing the (window)
nonadjacent radix$\tau$ expansions
for integers in all Euclidean imaginary quadratic number fields.
Keywords:algebraic integer, radix expression, window nonadjacent expansion, algorithm, point multiplication of elliptic curves, cryptography Categories:11A63, 11R04, 11Y16, 11Y40, 14G50 

5. CJM 2007 (vol 59 pp. 673)
 Ash, Avner; Friedberg, Solomon

Hecke $L$Functions and the Distribution of Totally Positive Integers
Let $K$ be a totally real number field of degree $n$. We show that
the number of totally positive integers
(or more generally the number of totally positive elements of a given fractional ideal)
of given trace is evenly distributed around its expected value, which is
obtained from geometric considerations.
This result depends on unfolding an integral over
a compact torus.
Keywords:Eisenstein series, toroidal integral, Fourier series, Hecke $L$function, totally positive integer, trace Categories:11M41, 11F30, , 11F55, 11H06, 11R47 

6. CJM 2005 (vol 57 pp. 1080)
 Pritsker, Igor E.

The GelfondSchnirelman Method in Prime Number Theory
The original GelfondSchnirelman method, proposed in 1936, uses
polynomials with integer coefficients and small norms on $[0,1]$
to give a Chebyshevtype lower bound in prime number theory. We
study a generalization of this method for polynomials in many
variables. Our main result is a lower bound for the integral of
Chebyshev's $\psi$function, expressed in terms of the weighted
capacity. This extends previous work of Nair and Chudnovsky, and
connects the subject to the potential theory with external fields
generated by polynomialtype weights. We also solve the
corresponding potential theoretic problem, by finding the extremal
measure and its support.
Keywords:distribution of prime numbers, polynomials, integer, coefficients, weighted transfinite diameter, weighted capacity, potentials Categories:11N05, 31A15, 11C08 

7. CJM 2005 (vol 57 pp. 648)
 Nevins, Monica

Branching Rules for Principal Series Representations of $SL(2)$ over a $p$adic Field
We explicitly describe the decomposition into irreducibles of
the restriction of the principal
series representations of $SL(2,k)$, for $k$ a $p$adic field,
to each of its two maximal compact subgroups (up to conjugacy).
We identify these irreducible subrepresentations in the
Kirillovtype classification
of Shalika. We go on to explicitly describe the decomposition
of the reducible principal series of $SL(2,k)$ in terms of the
restrictions of its irreducible constituents to a maximal compact
subgroup.
Keywords:representations of $p$adic groups, $p$adic integers, orbit method, $K$types Categories:20G25, 22E35, 20H25 

8. CJM 2003 (vol 55 pp. 711)
 Broughan, Kevin A.

Adic Topologies for the Rational Integers
A topology on $\mathbb{Z}$, which gives a nice proof that the
set of prime integers is infinite, is characterised and examined.
It is found to be homeomorphic to $\mathbb{Q}$, with a compact
completion homeomorphic to the Cantor set. It has a natural place
in a family of topologies on $\mathbb{Z}$, which includes the
$p$adics, and one in which the set of rational primes $\mathbb{P}$
is dense. Examples from number theory are given, including the
primes and squares, Fermat numbers, Fibonacci numbers and $k$free
numbers.
Keywords:$p$adic, metrizable, quasivaluation, topological ring,, completion, inverse limit, diophantine equation, prime integers,, Fermat numbers, Fibonacci numbers Categories:11B05, 11B25, 11B50, 13J10, 13B35 
