Finiteness property in Cantor real numeration systems

TL;DR

A class of alternate bases which satisfy the so-called finiteness property of Cantor real base numeration systems is provided and the proof uses rewriting rules on the language of expansion in the corresponding numeration system.

Abstract

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 $β_k$-transformations for a given sequence of real bases $B=(β_k)_{k\geq 1}$, $β_k>1$. 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.

Theorems & Definitions (50)

• Definition 2.1
• Lemma 2.1
• proof
• Theorem 2.2: CC21
• Proposition 2.3
• proof
• Corollary 2.4
• Theorem 3.1
• Theorem 3.2
• Example 3.3