2012/02/09 by Zuzana Masáková, Edita Pelantová, Masáková, Zuzana +1
Computer Science · Mathematics · #Computability, Logic, AI Algorithms #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #math.NT #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.1202.1948
arxiv created 2012/02/09 · openalex publication_date 2012/02/09 · arxiv updated 2012/02/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the question of pure periodicity of expansions in the negative base numeration system. In analogy of Akiyama's result for positive Pisot unit base β, we find a sufficient condition so that there exist an interval J containing the origin such that the (-β)-expansion of every rational number from J is purely periodic. We focus on the case of quadratic bases and demonstrate the following difference between the negative and positive bases: It is known that the finiteness property (\rm Fin(β)=\Z[β]) is not only sufficient, but also necessary in the case of positive quadratic and cubic bases. We show that \rm Fin(-β)=\Z[β] is not necessary in the case of negative bases.