Two applications of the spectrum of numbers

TL;DR

This paper investigates the properties of the spectrum of numbers in complex base systems, linking accumulation points to representations and automata recognizability, with applications to Pisot numbers and complex division algorithms.

Contribution

It establishes a connection between spectrum accumulation points, number representations, and automata recognizability, extending results to complex bases and digit sets.

Findings

Spectrum has an accumulation point iff zero has a rigid representation.

Recognizability of zero representations by automata is characterized by spectrum properties.

Complex division algorithms are feasible if and only if the spectrum has no accumulation point.

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.