There are two mistakes in the referred paper. One is ridiculous and one is significant. But none is serious.

We find an asymptotic formula for the number of rational points near planar curves. More precisely, if f:ℝ → ℝ is a sufficiently smooth function defined on the interval [η,ξ], then the number of rational points with denominator no larger than Q that lie within a δ-neighborhood of the graph of f is shown to be asymptotically equivalent to (ξ-η)δQ².

This paper introduces some methods to determine the simultaneous approximation constants of a class of well approximable numbers ${\zeta }_1,{\zeta }_2,...,{\zeta }_k$. The approach relies on results on the connection between the set of all $s$-adic expansions ($s\ge 2$) of ${\zeta }_1,{\zeta }_2,...,{\zeta }_k$ and their associated approximation constants. As an application, explicit construction of real numbers ${\zeta }_1,{\zeta }_2,...,{\zeta }_k$ with prescribed approximation properties are deduced and illustrated by Matlab plots.

In this paper, we extend the theory of simultaneous Diophantine approximation to infinite dimensions. Moreover, we discuss Dirichlet-type theorems in a very general framework and define what it means for such a theorem to be optimal. We show that optimality is implied by but does not imply the existence of badly approximable points.

Introduction. Soit θ un élément de ¹=ℝ/ℤ. Considérons la suite des multiples de θ, $x={\left(n\theta \right)}_{n\in \mathbb{N}}$. Pour tout n ∈ ℕ, ordonnons les n+1 premiers termes de cette suite, 0 = y₀ ≤ y₁ ≤...≤ yₙ ≤ 1 = pθ, p=0,...,n. La suite (y₀,...,yₙ) découpe l’intervalle [0,1] en n+1 intervalles qui ont au plus trois longueurs distinctes, la plus grande de ces longueurs étant la somme des deux autres. Cette propriété a été conjecturé par Steinhaus, elle est étroitement liée au développement en fraction continue de θ. On peut aussi...

In the problem of (simultaneous) Diophantine approximation in ${ℝ}^3$ (in the spirit of Hurwitz’s theorem), lower bounds for the critical determinant of the special three-dimensional body $$K_2:(y^2+z^2)(x^2+y^2+z^2)\le 1$$ play an important role; see [1], [6]. This article deals with estimates from below for the critical determinant $\Delta \left(K_c\right)$ of more general star bodies $$K_c:{(y^2+z^2)}^{c/2}(x^2+y^2+z^2)\le 1,$$ where $c$ is any positive constant. These are obtained by inscribing into $K_c$ either a double cone, or an ellipsoid, or a double paraboloid, depending on the size of $c$.

Let $\Theta =({\theta }_1,{\theta }_2,{\theta }_3)\in {ℝ}^3$. Suppose that $1,{\theta }_1,{\theta }_2,{\theta }_3$ are linearly independent over $\mathbb{Z}$. For Diophantine exponents $$\begin{array}{cc}\hfill \alpha \left(\Theta \right)& =sup\{\gamma >0:\underset{t\to +\infty }{lim\; sup}{t}^{\gamma }{\psi }_{\Theta }\left(t\right)<+\infty \},\hfill \\ \hfill \beta \left(\Theta \right)& =sup\{\gamma >0:\underset{t\to +\infty }{lim\; inf}{t}^{\gamma }{\psi }_{\Theta }\left(t\right)<+\infty \}\hfill \end{array}$$ we prove $$\beta \left(\Theta \right)\ge \frac{1}{2}\left(\frac{\alpha \left(\Theta \right)}{1-\alpha \left(\Theta \right)}+\sqrt{{\left(\frac{\alpha \left(\Theta \right)}{1-\alpha \left(\Theta \right)}\right)}^2+\frac{4\alpha \left(\Theta \right)}{1-\alpha \left(\Theta \right)}}\right)\alpha \left(\Theta \right).$$

Let $\xi$ be a real number and let $n$ be a positive integer. We define four exponents of Diophantine approximation, which complement the exponents $w_n\left(\xi \right)$ and $w_n^{*}\left(\xi \right)$ defined by Mahler and Koksma. We calculate their six values when $n=2$ and $\xi$ is a real number whose continued fraction expansion coincides with some Sturmian sequence of positive integers, up to the initial terms. In particular, we obtain the exact exponent of approximation to such a continued fraction $\xi$ by quadratic surds.

On montre que les exposants de Lyapunov de l’algorithme de Jacobi-Perron, en dimension $d$ quelconque, sont tous différents et que la somme des exposants extrêmes est strictement positive. En particulier, si $d=2$, le deuxième exposant est strictement négatif.