For a positive integer $n$ and a real number $\xi$, let ${\lambda }_n\left(\xi \right)$ denote the supremum of the real numbers $\lambda$ such that there are arbitrarily large positive integers $q$ such that $\left|\right|q\xi \left|\right|,\left|\right|q{\xi }^2\left|\right|,...,\left|\right|q{\xi }^n\left|\right|$ are all less than $q^{-\lambda }$. Here, $\left|\right|·\left|\right|$ denotes the distance to the nearest integer. We study the set of values taken by the function ${\lambda }_n$ and, more generally, we are concerned with the joint spectrum of $({\lambda }_1,...,{\lambda }_n,...)$. We further address several open problems.

We show in detail that the category of general Roth systems or the category of semi-stable systems of linear inequalities of slope zero is a neutral Tannakian category. On the way, we present a new proof of the semi-stability of the tensor product of semi-stable systems. The proof is based on a numerical criterion for a system of linear inequalities to be semi-stable.

In a classic paper, W.G. Spohn established the to-date sharpest estimates from below for the simultaneous Diophantine approximation constants for three and more real numbers. As a by-result of his method which used Blichfeldt’s Theorem and the calculus of variations, he derived a bound for the critical determinant of the star body $$\left|x_1\right|\left({\left|x_1\right|}^3+{\left|x_2\right|}^3+{\left|x_3\right|}^3\right)\le 1.$$ In this little note, after a brief exposition of the basics of the geometry of numbers and its significance for Diophantine approximation, this latter result is improved...

Let d be a positive integer and α a real algebraic number of degree d + 1. Set $\alpha ̲:=(\alpha ,\alpha ²,...,{\alpha }^d)$. It is well-known that $c\left(\alpha ̲\right):=limin{f}_{q\to \infty }{q}^{1/d}·\left|\right|q\alpha ̲\left|\right|>0$, where ||·|| denotes the distance to the nearest integer. Furthermore, $c\left(\alpha ̲\right)n^{-1/d}\le c\left(n\alpha ̲\right)\le nc\left(\alpha ̲\right)$ for any integer n ≥ 1. Our main result asserts that there exists a real number C, depending only on α, such that $c\left(n\alpha ̲\right)\le C{n}^{-1/d}$ for any integer n ≥ 1.

A point $({\xi }_1,{\xi }_2)$ with coordinates in a subfield of $ℝ$ of transcendence degree one over $\mathbb{Q}$, with $1,{\xi }_1,{\xi }_2$ linearly independent over $\mathbb{Q}$, may have a uniform exponent of approximation by elements of ${\mathbb{Q}}^2$ that is strictly larger than the lower bound $1/2$ given by Dirichlet’s box principle. This appeared as a surprise, in connection to work of Davenport and Schmidt, for points of the parabola $\{(\xi ,{\xi }^2);\xi \in ℝ\}$. The goal of this paper is to show that this phenomenon extends to all real conics defined over $\mathbb{Q}$, and that the largest exponent of...