Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Periodicity and pure periodicity in alternate base systems

Zuzana Masáková, Edita Pelantová

Abstract

We study the Cantor real base numeration system which is a common generalization of two positional systems, namely the Cantor system with a sequence of integer bases and the Rényi system with one real base. We focus on the so-called alternate base $B$ given by a purely periodic sequence of real numbers greater than 1. We answer an open question of Charlier et al. on the set of numbers with eventually periodic $B$-expansions. We also investigate for which bases all sufficiently small rationals have a purely periodic $B$-expansion.

Periodicity and pure periodicity in alternate base systems

Abstract

We study the Cantor real base numeration system which is a common generalization of two positional systems, namely the Cantor system with a sequence of integer bases and the Rényi system with one real base. We focus on the so-called alternate base given by a purely periodic sequence of real numbers greater than 1. We answer an open question of Charlier et al. on the set of numbers with eventually periodic -expansions. We also investigate for which bases all sufficiently small rationals have a purely periodic -expansion.
Paper Structure (7 sections, 15 theorems, 72 equations)

This paper contains 7 sections, 15 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['thm:SchmidtLepsi']}
  4. Proof of Theorem \ref{['thm:necessary']}
  5. Proof of Theorem \ref{['thm:sufficient']}
  6. A class of bases with (PP)
  7. Comments

Key Result

Theorem 1

Let $\boldsymbol{\mathcal{B}}=(\beta_1,\dots,\beta_{p})$ be an alternate base and set $\delta=\prod_{i=1}^p \beta_i$.

Theorems & Definitions (30)

  • Theorem 1: CCK23
  • Theorem A
  • Theorem 2: A98
  • Theorem B
  • Theorem C
  • Theorem 3: CC21
  • Definition 4
  • Lemma 5
  • proof
  • Lemma 6
  • ...and 20 more