Combinatorical and Algebraic Structures Seminar
Session details
| Date: | 26.2.2008 |
| Speaker: | Edita Pelantová, FJFI, České vysoké učení technické |
| Title: | Asymptotic behaviour of beta-integers |
| Abstract: | Parry numbers give rise to $\beta$-integers which realize only a finite number of distances between consecutive elements and are thus in a certain sense close to ordinary integers. We will illustrate the similarity between sets $\mathbb N$ and $\mathbb Z^{+}_\beta=\{b_n \mid n \in \mathbb N\}$ for $\beta$ being a~Parry number by proving two properties: \begin{enumerate} \item We will show that $c_{\beta}=\lim_{n \to \infty}\dfrac{b_n}{n}$ exists and we will provide a~simple formula for $c_{\beta}$. \item For $\beta$ being moreover a~PV number with mutually distinct roots of its Parry polynomial, we will prove that $(b_n-c_{\beta}\,n)_{n \in \mathbb N}$ is a~bounded sequence. \end{enumerate} |
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