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Acta Mathematica Hungarica

, Volume 156, Issue 2, pp 391–407 | Cite as

Two applications of the spectrum of numbers

  • Ch. FrougnyEmail author
  • E. Pelantová
Article
  • 32 Downloads

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.

Key words and phrases

spectrum Pisot number Büchi automaton 

Mathematics Subject Classification

11K16 68Q45 

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Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2018

Authors and Affiliations

  1. 1.IRIF, UMR 8243 CNRS and Université Paris-DiderotParis-DiderotFrance
  2. 2.Doppler Institute for Mathematical Physics and Applied Mathematics, and Department of MathematicsFNSPE, Czech Technical University in PraguePragueCzech Republic

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