2023/02/21 by Zuzana Masáková, Edita Pelantová, Masáková, Zuzana +3
Computer Science · Mathematics · #Combinatorics (math.CO) #Computability, Logic, AI Algorithms #Discrete Mathematics (cs.DM) #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2302.10708
openalex publication_date 2023/02/21 · openalex created_date 2023/02/24 · openalex updated_date 2026/07/28
For alternate Cantor real base numeration systems we generalize the result of Frougny and~Solomyak on~arithmetics on the set of numbers with finite expansion. We provide a class of alternate bases which satisfy the so-called finiteness property. The proof uses rewriting rules on the~language of~expansions in the corresponding numeration system. The proof is constructive and provides a~method for~performing addition of~expansions in Cantor real bases. We consider a numeration system which is a common generalization of the positional systems introduced by Cantor and Rényi. Number representations are obtained using a composition of βk-transformations for a given sequence of real bases B=(βk)k≥ 1, βk>1. We focus on~arithmetical properties of the set of numbers with finite B-expansion in case that B is an alternate base, i.e. B is a periodic sequence. We provide necessary conditions for the so-called finiteness property. We further show a~sufficient condition using rewriting rules on the~language of~representations. The proof is constructive and provides a~method for~performing addition of~expansions in alternate bases. Finally, we give a family of alternate bases that satisfy this sufficient condition. Our work generalizes the results of Frougny and Solomyak obtained for the case when the base B is a constant sequence.