In recent years, starting with the paper [B-D-S], we have investigated the possibility of characterizing countable subgroups of the torus $T=\mathbf{R}/\mathbf{Z}$ by subsets of $\mathbf{Z}$. Here we consider new types of subgroups: let $K\subseteq T$ be a Kronecker set (a compact set on which every continuous function $f:K\to T$ can be uniformly approximated by characters of $T$), and $G$ the group generated by $K$. We prove (Theorem 1) that $G$ can be characterized by a subset of ${\mathbf{Z}}^2$ (instead of a subset of $\mathbf{Z}$). If $K$ is finite, Theorem 1 implies our earlier result...

Let $L(\theta ,\lambda )$ be the set of limit points of the fractional parts $\left\{\lambda {\theta }^n\right\}$, $n=0,1,2,\cdots$, where $\theta$ is a Pisot number and $\lambda \in \mathbb{Q}\left(\theta \right)$. Using a description of $L(\theta ,\lambda )$, due to Dubickas, we show that there is a sequence ${\left({\lambda }_n\right)}_{n\ge 0}$ of elements of $\mathbb{Q}\left(\theta \right)$ such that $Card\left(L(\theta ,{\lambda }_n)\right)<Card\left(L(\theta ,{\lambda }_{n+1})\right)$, $\forall$$n\ge 0$. Also, we prove that the...

Consider the group ${SL}_2\left({\mathbf{O}}_K\right)$ over the ring of algebraic integers of a number field $K$. Define the height of a matrix to be the maximum over all the conjugates of its entries in absolute value. Let ${SL}_2({\mathbf{O}}_K,t)$ be the number of matrices in ${SL}_2\left({\mathbf{O}}_K\right)$ with height bounded by $t$. We determine the asymptotic behaviour of ${SL}_2({\mathbf{O}}_K,t)$ as $t$ goes to infinity including an error term,$${SL}_2({\mathbf{O}}_K,t)=C{t}^{2n}+O\left(t^{2n-\eta }\right)$$with $n$ being the degree of $K$. The constant $C$ involves the discriminant of $K$, an integral depending only on the signature of $K$, and the value of the Dedekind zeta function...