Two applications of the spectrum of numbers

Abstract

Let the base $β$ be a complex number, $|β|>1$, and let $A \subset \C$ be a finite alphabet of digits. The \emph{$A$-spectrum} of $β$ is the set $S_{A}(β) = \{\sum_{k=0}^n a_kβ^k \mid n \in \mathbb{N}, \ a_k \in {A}\}$. We show that the spectrum $S_{A}(β)$ has an accumulation point if and only if $0$ has a particular $(β, A)$-representation, said to be \emph{rigid}. The first application is restricted to the case that $β>1 $ and the alphabet is $A=\{-M, \ldots, M\}$, $M \ge 1$ integer. We show that the set $Z_{β,M}$ of infinite $(β, A)$-representations of $0$ is recognizable by a finite Büchi automaton if and only if the spectrum $S_A(β)$ has no accumulation point. Using a result of Akiyama-Komornik and Feng, this implies that $Z_{β, M}$ is recognizable by a finite Büchi automaton for any positive integer $M \ge \lceil β\rceil -1$ if and only if $β$ is a Pisot number. This improves the previous bound $M \ge \lceil β\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 $(β,A)$-representation of the divisor can be small. The numeration system $(β,A)$ is said to \emph{allow preprocessing} if there exists a finite list of transformations on the divisor which achieve this task. We show that $(β,A )$ allows preprocessing if and only if the spectrum $S_{A}(β)$ has no accumulation point.