2023/12/21 by Charlier, Emilie, Cisternino, Célia, Masáková, Zuzana +1
#11K16 #11R06 #37B10 #68Q45 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2312.13767
We consider Cantor real numeration system as a frame in which every non-negative real number has a positional representation. The system is defined using a bi-infinite sequence \Beta=(βn)n∈\Z of real numbers greater than one. We introduce the set of \Beta-integers and code the sequence of gaps between consecutive \Beta-integers by a symbolic sequence in general over the alphabet \N. We show that this sequence is S-adic. We focus on alternate base systems, where the sequence \Beta of bases is periodic and characterize alternate bases \Beta, in which \Beta-integers can be coded using a symbolic sequence v\Beta over a finite alphabet. With these so-called Parry alternate bases we associate some substitutions and show that v_\Beta is a fixed point of their composition. The paper generalizes results of Fabre and Burdík et al. obtained for the Rényi numerations systems, i.e., in the case when the Cantor base \Beta is a constant sequence.