Two applications of the spectrum of numbers

Abstract

Let the base ${\beta}$ be a complex number, ${|\beta| > 1}$, and let ${A \subset \mathbb{C}}$ be a finite alphabet of digits. The A-spectrum of ${\beta}$ is the set ${{S}_{A}(\beta) = {\{\Sigma^{n}_{k=0} {a}_{k}\beta^{k} | n\in \mathbb{N}, {a}_{k} \in A\}}}$. We show that the spectrum ${{S}_{A}(\beta)}$ has an accumulation point if and only if 0 has a particular ${(\beta, A)}$-representation, said to be rigid.

The first application is restricted to the case that ${\beta > 1}$ and the alphabet is A = {−M, . . . , M}, ${{M \geq}}$1 integer. We show that the set ${{Z}_{\beta, M}}$ of infinite ${(\beta, A)}$-representations of 0 is recognizable by a finite Büchi automaton if and only if the spectrum ${{S}_{A}(\beta)}$ has no accumulation point. Using a result of Akiyama–Komornik and Feng, this implies that ${{Z}_{\beta, M}}$ is recognizable by a finite Büchi automaton for any positive integer ${M \geq\lceil {\beta\rceil-1}}$ if and only if ${{\beta}}$ is a Pisot number. This improves the previous bound ${M \geq \lceil \beta\rceil}$.

For the second application the base and the digits are complex. We consider the on-line algorithm for division of Trivedi and Ercegovac generalized to a complex numeration system. In on-line arithmetic the operands and results are processed in a digit serial manner, starting with the most significant digit. The divisor must be far from 0, which means that no prefix of the ${(\beta,A)}$-representation of the divisor can be small. The numeration system ${(\beta,A)}$ is said to allow preprocessing if there exists a finite list of transformations on the divisor which achieve this task. We show that ${(\beta,A)}$ allows preprocessing if and only if the spectrum ${{S}_{A}(\beta)}$ has no accumulation point.