Two applications of the spectrum of numbers.

Let the base β be a complex number, |β|>1, and let A⊂C be a finite alphabet of digits. The A-spectrum of β is the set SA(β)={Σk=0nakβk|n∈N,ak∈A}. We show that the spectrum SA(β) has an accumulation point if and only if 0 has a particular (β,A)-representation, said to be rigid.The first application is restricted to the case that β>1 and the alphabet is A = {−M,..., M}, M≥1 integer. We show that the set Zβ,M of infinite (β,A)-representations of 0 is recognizable by a finite Büchi automaton if and only if the spectrum SA(β) has no accumulation point. Using a result of Akiyama-Komornik and Feng, this implies that Zβ,M is recognizable by a finite Büchi automaton for any positive integer M≥⌈β⌉-1 if and only if β is a Pisot number. This improves the previous bound M≥⌈β⌉.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 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 SA(β) has no accumulation point. [ABSTRACT FROM AUTHOR]