The notion of designs in Grassmannian spaces was introduced by the author and R. Coulangeon, G. Nebe, in [3]. After having recalled some basic properties of these objects and the connections with the theory of lattices, we prove that the sequence of Barnes-Wall lattices hold $6$-Grassmannian designs. We also discuss the connections between the notion of Grassmannian design and the notion of design associated with the symmetric space of the totally isotropic subspaces in a binary quadratic space, which...

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.

Let two lattices ${\Lambda }^{\text{'}},{\Lambda }^{\text{'}\text{'}}\subset {ℝ}^s$ have the same number of points on each hyperbolic surface $|x₁...x_s|=C$. We investigate the case when Λ’, Λ” are sublattices of ${\mathbb{Z}}^s$ of the same prime index and show that then Λ’ and Λ” must coincide up to renumbering the coordinate axes and changing their directions.