Two applications of the spectrum of numbers

Abstract

Let the base $\beta$ be a complex number, $|\beta|>1$, and let $A \subset \C$ be a finite alphabet of digits. The \emph{$A$-spectrum} of $\beta$ is the set $S_{A}(\beta) = \{\sum_{k=0}^n a_k\beta^k \mid n \in \mathbb{N}, \ a_k \in {A}\}$. 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 \emph{rigid}. The first application is restricted to the case that $\beta >1 $ and the alphabet is $A=\{-M, \ldots, M\}$, $M \ge 1$ integer. We show that the set $Z_{\beta,M}$ of infinite $(\beta, A)$-representations of $0$ is recognizable by a finite B\"uchi 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 B\"uchi automaton for any positive integer $M \ge \lceil \beta \rceil -1$ if and only if $\beta$ is a Pisot number. This improves the previous bound $M \ge \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 \emph{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.