For $\mathbf{r}=(r_0,...,r_{d-1})\in {ℝ}^d$ define the function$${\tau }_{\mathbf{r}}:{\mathbb{Z}}^d\to {\mathbb{Z}}^d,\mathbf{z}=(z_0,...,z_{d-1})\mapsto (z_1,...,z_{d-1},-⌊\mathbf{rz}⌋),$$where $\mathbf{rz}$ is the scalar product of the vectors $\mathbf{r}$ and $\mathbf{z}$. If each orbit of ${\tau }_{\mathbf{r}}$ ends up at $\mathbf{0}$, we call ${\tau }_{\mathbf{r}}$ a shift radix system. It is a well-known fact that each orbit of ${\tau }_{\mathbf{r}}$ ends up periodically if the polynomial $t^d+r_{d-1}{t}^{d-1}+\cdots +r_0$ associated to $\mathbf{r}$ is contractive. On the other hand, whenever this polynomial has at least one root outside the unit disc, there exist starting vectors that give rise to unbounded orbits. The present paper deals with the remaining situations of periodicity properties of...

In a recent work we gave some estimations for exponential sums of the form ${\sum }_{n\le x}\Lambda \left(n\right)exp\left(2i\pi (f\left(n\right)+\beta n)\right)$, where Λ denotes the von Mangoldt function, f a digital function, and β a real parameter. The aim of this work is to show how these results can be used to study the statistical properties of digital functions along prime numbers.