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Parry condition, existence and uniqueness of alternate bases

Émilie Charlier, Savinien Kreczman, Zuzana Masáková, Edita Pelantová

Abstract

Alternate bases are a numeration system that generalizes the Rényi numeration system. It is common in this context to construct examples or counter-examples by specifying the expansions of $1$ in the desired system. While it is easy to show when a system with given expansions of $1$ exists in the Rényi case, the same is not true in the alternate case. In this article, we establish conditions for given words to be the expansions of $1$ in the alternate case. To do so, we use a fixed point theorem on matrices defined from the expansions and obtain the elements of the base from the components of the fixed point. We also obtain a partial result for the uniqueness of such a base. In the latter parts of the article, we use similar techniques to prove the existence of bases with a given sequence of $B$-integers.

Parry condition, existence and uniqueness of alternate bases

Abstract

Alternate bases are a numeration system that generalizes the Rényi numeration system. It is common in this context to construct examples or counter-examples by specifying the expansions of in the desired system. While it is easy to show when a system with given expansions of exists in the Rényi case, the same is not true in the alternate case. In this article, we establish conditions for given words to be the expansions of in the alternate case. To do so, we use a fixed point theorem on matrices defined from the expansions and obtain the elements of the base from the components of the fixed point. We also obtain a partial result for the uniqueness of such a base. In the latter parts of the article, we use similar techniques to prove the existence of bases with a given sequence of -integers.
Paper Structure (10 sections, 24 theorems, 81 equations)

This paper contains 10 sections, 24 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Parry conditions characterize the greedy and the quasi-greedy expansions of 1
  4. Uniqueness of the alternate base with prescribed greedy or quasi-greedy expansions of 1
  5. Consequences of a theorem of Furstenberg
  6. Existence of an alternate base with given representations of 1
  7. A family of sequences faithfully coding B-integers
  8. Arnoux-Rauzy sequences
  9. Sequences associated with N-continued fractions
  10. Comments

Key Result

Lemma 1

Let $B=(\beta_n)_{n\in \mathbb Z}$ be a Cantor real base. A sequence $(a_n)_{n\in\mathbb Z}$ of non-negative integers, with $a_n=0$ for sufficiently large positive $n$, is the $B$-expansion of some non-negative real number if and only if we have and

Theorems & Definitions (41)

  • Lemma 1
  • Theorem 2: CharlierCisterninoMasakovaPelantova:2025
  • Corollary 3
  • proof
  • Lemma 4
  • Proposition 5
  • proof
  • Corollary 6
  • Corollary 7
  • Proposition 8
  • ...and 31 more