1. CJM 2009 (vol 61 pp. 165)
 Laurent, Michel

Exponents of Diophantine Approximation in Dimension Two
Let $\Theta=(\alpha,\beta)$ be a point in $\bR^2$, with $1,\alpha,
\beta$ linearly independent over $\bQ$. We attach to $\Theta$ a
quadruple $\Omega(\Theta)$ of exponents that measure the quality
of approximation to $\Theta$ both by rational points and by
rational lines. The two ``uniform'' components of $\Omega(\Theta)$
are related by an equation due to Jarn\'\i k, and the four
exponents satisfy two inequalities that refine Khintchine's
transference principle. Conversely, we show that for any quadruple
$\Omega$ fulfilling these necessary conditions, there exists
a point $\Theta\in \bR^2$ for which $\Omega(\Theta) =\Omega$.
Categories:11J13, 11J70 

2. CJM 2007 (vol 59 pp. 503)
 Chevallier, Nicolas

Cyclic Groups and the Three Distance Theorem
We give a two dimensional extension of the three distance Theorem. Let
$\theta$ be in $\mathbf{R}^{2}$ and let $q$ be in $\mathbf{N}$. There exists a
triangulation of $\mathbf{R}^{2}$ invariant by $\mathbf{Z}^{2}$translations,
whose set of vertices is $\mathbf{Z}^{2}+\{0,\theta,\dots,q\theta\}$, and whose
number of different triangles, up to translations, is bounded above by a
constant which does not depend on $\theta$ and $q$.
Categories:11J70, 11J71, 11J13 

3. CJM 2007 (vol 59 pp. 211)
 Roy, Damien

On Two Exponents of Approximation Related to a Real Number and Its Square
For each real number $\xi$, let $\lambdahat_2(\xi)$ denote the
supremum of all real numbers $\lambda$ such that, for each
sufficiently large $X$, the inequalities $x_0 \le X$,
$x_0\xix_1 \le X^{\lambda}$ and $x_0\xi^2x_2 \le
X^{\lambda}$ admit a solution in integers $x_0$, $x_1$ and $x_2$
not all zero, and let $\omegahat_2(\xi)$ denote the supremum of
all real numbers $\omega$ such that, for each sufficiently large
$X$, the dual inequalities $x_0+x_1\xi+x_2\xi^2 \le
X^{\omega}$, $x_1 \le X$ and $x_2 \le X$ admit a solution in
integers $x_0$, $x_1$ and $x_2$ not all zero. Answering a
question of Y.~Bugeaud and M.~Laurent, we show that the exponents
$\lambdahat_2(\xi)$ where $\xi$ ranges through all real numbers
with $[\bQ(\xi)\wcol\bQ]>2$ form a dense subset of the interval $[1/2,
(\sqrt{5}1)/2]$ while, for the same values of $\xi$, the dual
exponents $\omegahat_2(\xi)$ form a dense subset of $[2,
(\sqrt{5}+3)/2]$. Part of the proof rests on a result of
V.~Jarn\'{\i}k showing that $\lambdahat_2(\xi) =
1\omegahat_2(\xi)^{1}$ for any real number $\xi$ with
$[\bQ(\xi)\wcol\bQ]>2$.
Categories:11J13, 11J82 

4. CJM 2002 (vol 54 pp. 1305)