1. CJM 2012 (vol 65 pp. 222)
 Sauer, N. W.

Distance Sets of Urysohn Metric Spaces
A metric space $\mathrm{M}=(M;\operatorname{d})$ is {\em homogeneous} if for every
isometry $f$ of a finite subspace of $\mathrm{M}$ to a subspace of
$\mathrm{M}$ there exists an isometry of $\mathrm{M}$ onto
$\mathrm{M}$ extending $f$. The space $\mathrm{M}$ is {\em universal}
if it isometrically embeds every finite metric space $\mathrm{F}$ with
$\operatorname{dist}(\mathrm{F})\subseteq \operatorname{dist}(\mathrm{M})$. (With
$\operatorname{dist}(\mathrm{M})$ being the set of distances between points in
$\mathrm{M}$.)
A metric space $\boldsymbol{U}$ is an {\em Urysohn} metric space if
it is homogeneous, universal, separable and complete. (It is not
difficult to deduce
that an Urysohn metric space $\boldsymbol{U}$ isometrically embeds
every separable metric space $\mathrm{M}$ with
$\operatorname{dist}(\mathrm{M})\subseteq \operatorname{dist}(\boldsymbol{U})$.)
The main results are: (1) A characterization of the sets
$\operatorname{dist}(\boldsymbol{U})$ for Urysohn metric spaces $\boldsymbol{U}$.
(2) If $R$ is the distance set of an Urysohn metric space and
$\mathrm{M}$ and $\mathrm{N}$ are two metric spaces, of any
cardinality with distances in $R$, then they amalgamate disjointly to
a metric space with distances in $R$. (3) The completion of every
homogeneous, universal, separable metric space $\mathrm{M}$ is
homogeneous.
Keywords:partitions of metric spaces, Ramsey theory, metric geometry, Urysohn metric space, oscillation stability Categories:03E02, 22F05, 05C55, 05D10, 22A05, 51F99 

2. CJM 2011 (vol 64 pp. 1201)
 Aistleitner, Christoph; Elsholtz, Christian

The Central Limit Theorem for Subsequences in Probabilistic Number Theory
Let $(n_k)_{k \geq 1}$ be an increasing sequence of positive integers, and let $f(x)$ be a real function satisfying
\begin{equation}
\tag{1}
f(x+1)=f(x), \qquad \int_0^1 f(x) ~dx=0,\qquad
\operatorname{Var_{[0,1]}}
f \lt \infty.
\end{equation}
If $\lim_{k \to \infty} \frac{n_{k+1}}{n_k} = \infty$
the distribution of
\begin{equation}
\tag{2}
\frac{\sum_{k=1}^N f(n_k x)}{\sqrt{N}}
\end{equation}
converges to a Gaussian distribution. In the case
$$
1 \lt \liminf_{k \to \infty} \frac{n_{k+1}}{n_k}, \qquad \limsup_{k \to \infty} \frac{n_{k+1}}{n_k} \lt \infty
$$
there is a complex interplay between the analytic properties of the
function $f$, the numbertheoretic properties of $(n_k)_{k \geq 1}$,
and the limit distribution of (2).
In this paper we prove that any sequence $(n_k)_{k \geq 1}$ satisfying
$\limsup_{k \to \infty} \frac{n_{k+1}}{n_k} = 1$ contains a nontrivial
subsequence $(m_k)_{k \geq 1}$ such that for any function satisfying
(1) the distribution of
$$
\frac{\sum_{k=1}^N f(m_k x)}{\sqrt{N}}
$$
converges to a Gaussian distribution. This result is best possible: for any
$\varepsilon\gt 0$ there exists a sequence $(n_k)_{k \geq 1}$ satisfying $\limsup_{k \to
\infty} \frac{n_{k+1}}{n_k} \leq 1 + \varepsilon$ such that for every nontrivial
subsequence $(m_k)_{k \geq 1}$ of $(n_k)_{k \geq 1}$ the distribution
of (2) does not converge to a Gaussian distribution for some $f$.
Our result can be viewed as a Ramsey type result: a sufficiently dense
increasing integer sequence contains a subsequence having a certain
requested numbertheoretic property.
Keywords:central limit theorem, lacunary sequences, linear Diophantine equations, Ramsey type theorem Categories:60F05, 42A55, 11D04, 05C55, 11K06 
