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Séminaire Lotharingien de Combinatoire [electronic only]

A note on alternating sums.

The Electronic Journal of Combinatorics [electronic only]

Approximate counting : an alternative approach

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

Finite and periodic orbits of shift radix systems

Journal de Théorie des Nombres de Bordeaux

For $\mathbf{r}=\left({r}_{0},...,{r}_{d-1}\right)\in {ℝ}^{d}$ define the function ${\tau }_{\mathbf{r}}:\phantom{\rule{0.166667em}{0ex}}{ℤ}^{d}\to {ℤ}^{d},\phantom{\rule{1em}{0ex}}\mathbf{z}=\left({z}_{0},...,{z}_{d-1}\right)↦\left({z}_{1},...,{z}_{d-1},-⌊\mathbf{rz}⌋\right),$ 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...

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