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 Buchi 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 Buchi 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.