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Transcendence for Pisot Morphic Words over an Algebraic Base

Pavol Kebis, Florian Luca, Joel Ouaknine, Andrew Scoones, James Worrell

TL;DR

The paper studies bases $\beta$ with $|\beta|>1$ for which the base-$\beta$ expansion of a morphic word is either in $\mathbb{Q}(\beta)$ or transcendental. It introduces the echoing concept to connect combinatorial structure of morphic words with a rational-transcendental dichotomy, and uses a Subspace Theorem framework to prove the main dichotomy for echoing words. It then strengthens the condition to 'strongly echoing' to obtain outright transcendence, and applies this to the Tribonacci and general $k$-bonacci words, establishing strong transcendence results. Assuming the Pisot conjecture, the results extend to irreducible Pisot morphisms over arbitrary alphabets, yielding a broad algebraic-base Cobham-type transcendence theory for morphic sequences.

Abstract

It is known that for a uniform morphic sequence $\boldsymbol u = \langle u_n\rangle_{n=0}^\infty$ and an algebraic number $β$ such that $|β|>1$, the number $[\![\boldsymbol{u} ]\!]_β:=\sum_{n=0}^\infty \frac{u_n}{β^n}$ either lies in $\mathbb Q(β)$ or is transcendental. In this paper we show a similar rational-transcendental dichotomy for sequences defined by irreducible Pisot morphisms. Subject to the Pisot conjecture (an irreducible Pisot morphism has pure discrete spectrum), we generalise the latter result to arbitrary finite alphabets. In certain cases we are able to show transcendence of $[\![\boldsymbol{u}]\!]_β$ outright. In particular, for $k\geq 2$, if $\boldsymbol u$ is the $k$-bonacci word then $[\![\boldsymbol{u}]\!]_β$ is transcendental.

Transcendence for Pisot Morphic Words over an Algebraic Base

TL;DR

The paper studies bases with for which the base- expansion of a morphic word is either in or transcendental. It introduces the echoing concept to connect combinatorial structure of morphic words with a rational-transcendental dichotomy, and uses a Subspace Theorem framework to prove the main dichotomy for echoing words. It then strengthens the condition to 'strongly echoing' to obtain outright transcendence, and applies this to the Tribonacci and general -bonacci words, establishing strong transcendence results. Assuming the Pisot conjecture, the results extend to irreducible Pisot morphisms over arbitrary alphabets, yielding a broad algebraic-base Cobham-type transcendence theory for morphic sequences.

Abstract

It is known that for a uniform morphic sequence and an algebraic number such that , the number either lies in or is transcendental. In this paper we show a similar rational-transcendental dichotomy for sequences defined by irreducible Pisot morphisms. Subject to the Pisot conjecture (an irreducible Pisot morphism has pure discrete spectrum), we generalise the latter result to arbitrary finite alphabets. In certain cases we are able to show transcendence of outright. In particular, for , if is the -bonacci word then is transcendental.
Paper Structure (21 sections, 12 theorems, 53 equations, 1 figure)

This paper contains 21 sections, 12 theorems, 53 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Morphic Sequences
  4. Number Theory
  5. Echoing Words
  6. The Fibonacci Word
  7. The Tribonacci Word
  8. Definition of Echoing Words
  9. Binary Pisot Morphic Words Are Echoing
  10. Balanced Pairs
  11. The Tribonacci Word is Echoing
  12. Pisot Morphic Words are Echoing
  13. Transcendence for Echoing Words
  14. Strongly Echoing Sequences
  15. The Non-Vanishing Condition
  16. ...and 6 more sections

Key Result

Theorem 1

Let $S \subseteq M(K)$ be a finite set of places of $K$ that contains all Archimedean places. Let $v_0 \in S$ be a distinguished place and choose a continuation of $|\cdot |_{v_0}$ to $\overline{\mathbb Q}$, also denoted $|\cdot |_{v_0}$. Given $m\geq 2$, let $L(x_1,\ldots,x_{m})$ be a linear form w is contained in a finite union of proper linear subspaces of $K^m$.

Figures (1)

  • Figure 1: An alternating cycle for a cycle cover $\mathcal{C}$. The blue edges lie in $\mathcal{C}$ and the black edges are not in $\mathcal{C}$. Removing the blue edges from $\mathcal{C}$ and replacing them by the black edges yields a new cycle cover. On the other hand, if $\mathcal{C}'$ is a cycle cover other than $\mathcal{C}$ then the symmetric difference of the respective sets of edges in $\mathcal{C}$ and $\mathcal{C}'$ can be partitioned into alternating paths of the above form

Theorems & Definitions (20)

  • Theorem 1
  • Proposition 2
  • Proposition 3
  • Definition 4
  • Theorem 5: Hollander and Solomayak
  • Theorem 6: Barge and Diamond
  • Theorem 7
  • Lemma 8
  • proof
  • Theorem 9
  • ...and 10 more