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Mahler equations for Zeckendorf numeration

Olivier Carton, Reem Yassawi

TL;DR

The paper generalizes Christol’s theorem to Zeckendorf numeration by introducing Z-Mahler equations and the notion of Z-regular sequences, establishing a tight correspondence with weighted automata. It provides a constructive framework: isolating Z-Mahler equations yield Z-regular series computable by reading Zeckendorf representations, and conversely, Z-regular series satisfy Z-Mahler equations; the authors introduce a universal Z-automaton and derive state bounds. The work extends Becker–Dumas results to Zeckendorf numeration and demonstrates both directions of the automata–Mahler correspondence, including a non-regular Z-Mahler example to show the necessity of isolation. It also indicates possible generalizations to Pisot-based numeration systems and offers a new automata-based approach to generating classical $q$-regular sequences, enriching the interaction between automata theory and generalized Mahler equations.

Abstract

We define generalised equations of Z-Mahler type, based on the Zeckendorf numeration system. We show that if a sequence over a commutative ring is Z-regular, then it is the sequence of coefficients of a series which is a solution of a Z-Mahler equation. Conversely, if the Z-Mahler equation is isolating, then its solutions define Z-regular sequences. This is a generalisation of results of Becker and Dumas. We provide an example to show that there exist non-isolating Z-Mahler equations whose solutions do not define Z-regular sequences. Our proof yields a new construction of weighted automata that generate classical q-regular sequences.

Mahler equations for Zeckendorf numeration

TL;DR

The paper generalizes Christol’s theorem to Zeckendorf numeration by introducing Z-Mahler equations and the notion of Z-regular sequences, establishing a tight correspondence with weighted automata. It provides a constructive framework: isolating Z-Mahler equations yield Z-regular series computable by reading Zeckendorf representations, and conversely, Z-regular series satisfy Z-Mahler equations; the authors introduce a universal Z-automaton and derive state bounds. The work extends Becker–Dumas results to Zeckendorf numeration and demonstrates both directions of the automata–Mahler correspondence, including a non-regular Z-Mahler example to show the necessity of isolation. It also indicates possible generalizations to Pisot-based numeration systems and offers a new automata-based approach to generating classical -regular sequences, enriching the interaction between automata theory and generalized Mahler equations.

Abstract

We define generalised equations of Z-Mahler type, based on the Zeckendorf numeration system. We show that if a sequence over a commutative ring is Z-regular, then it is the sequence of coefficients of a series which is a solution of a Z-Mahler equation. Conversely, if the Z-Mahler equation is isolating, then its solutions define Z-regular sequences. This is a generalisation of results of Becker and Dumas. We provide an example to show that there exist non-isolating Z-Mahler equations whose solutions do not define Z-regular sequences. Our proof yields a new construction of weighted automata that generate classical q-regular sequences.
Paper Structure (20 sections, 31 theorems, 77 equations, 6 figures)

This paper contains 20 sections, 31 theorems, 77 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Mahler equations and weighted automata
  3. Basic notation
  4. Automata
  5. Matrix representations of weighted automata
  6. From weighted automata to automatic sequences
  7. Robustness of weighted automata
  8. $q$-Mahler equations and weighted automata
  9. From weighted automata to $q$-Mahler equations
  10. Reformulating isolating $q$-Mahler equations
  11. From isolating $q$-Mahler equations to weighted automata
  12. Mahler equations for Zeckendorf numeration
  13. The Zeckendorf numeration
  14. The function $ϕ$ and its almost-linearity
  15. Regularity of the linearity defect
  16. ...and 5 more sections

Key Result

Theorem 1

Let $P$ be an isolating $q$-Mahler equation over the commutative ring $R$. Let $f(x) = ∑_{n⩾ 0}f_n x^n$ satisfy $P(x, f(x))=0$. Then the weighted automaton $𝒜 = 𝒜_{P,f_0}$ generates $f$.

Figures (6)

  • Figure 1: A weighted automaton that generates the Thue-Morse sequence $(a_n)$, where $a_n= \operatorname{weight}_{𝒜}((n)_2)$. The weight of an edge is given in red, and the blue numbers are the digits we read in $(n)_2$.
  • Figure 2: A automaton recognising addition base-$2$. A string over $\{0,1\}^3$, whose letters here are written as column vectors $xyz$, is accepted in direct reading if and only if it equals $(m)_2 \otimes (n)_2 \otimes (m+n)_2$.
  • Figure 3: The weighted automaton for $f(x) = (α_{1,0}+α_{1,1}x+α_{1,2}x^2+α_{1,3}x^3)f(x^2)$
  • Figure 4: The automaton for a 2-Mahler equation of exponent 2 and height 3.
  • Figure 5: An automaton computing $δ(m-n,n)$, given $(m)_Z ⊟ (n)_Z$.
  • ...and 1 more figures

Theorems & Definitions (62)

  • Theorem 1
  • Theorem 2
  • Example 3
  • Example 4
  • Lemma 5
  • Proposition 6
  • proof
  • Corollary 7
  • Theorem 8
  • proof
  • ...and 52 more