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Symbols frequencies in the Thue--Morse word in base $3/2$ and related conjectures

Julien Cassaigne, Bastiàn Espinoza, Michel Rigo, Manon Stipulanti

Abstract

We study a binary Thue--Morse-type sequence arising from the base-$3/2$ expansion of integers, an archetypal automatic sequence in a rational base numeration system. Because the sequence is generated by a periodic iteration of morphisms rather than a single primitive substitution, classical Perron--Frobenius methods do not directly apply to determine symbol frequencies. We prove that both symbols ${\tt 0},{\tt 1}$ occur with frequency $1/2$ and we show uniform recurrence and symmetry properties of its set of factors. The proof reveals a structural bridge between combinatorics on words and harmonic analysis: the first difference sequence is shown to be Toeplitz, providing dynamical rigidity, while filtered frequencies naturally encode a dyadic structure that lifts to the compact group of $2$-adic integers. In this $2$-adic setting, desubstitution becomes a linear operator on Fourier coefficients, and a spectral contraction argument enforces uniqueness of limiting densities. Our results answer several conjectures of Dekking (on a sibling sequence) and illustrate how harmonic analysis on compact groups can be fruitfully combined with substitution dynamics.

Symbols frequencies in the Thue--Morse word in base $3/2$ and related conjectures

Abstract

We study a binary Thue--Morse-type sequence arising from the base- expansion of integers, an archetypal automatic sequence in a rational base numeration system. Because the sequence is generated by a periodic iteration of morphisms rather than a single primitive substitution, classical Perron--Frobenius methods do not directly apply to determine symbol frequencies. We prove that both symbols occur with frequency and we show uniform recurrence and symmetry properties of its set of factors. The proof reveals a structural bridge between combinatorics on words and harmonic analysis: the first difference sequence is shown to be Toeplitz, providing dynamical rigidity, while filtered frequencies naturally encode a dyadic structure that lifts to the compact group of -adic integers. In this -adic setting, desubstitution becomes a linear operator on Fourier coefficients, and a spectral contraction argument enforces uniqueness of limiting densities. Our results answer several conjectures of Dekking (on a sibling sequence) and illustrate how harmonic analysis on compact groups can be fruitfully combined with substitution dynamics.
Paper Structure (22 sections, 18 theorems, 137 equations, 7 figures, 1 table)

This paper contains 22 sections, 18 theorems, 137 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Substitutions or iterated morphisms
  3. Periodically iterated morphisms
  4. The base $3/2$ and the corresponding Thue--Morse word
  5. Dekking's variation
  6. Organization of the paper and our contributions
  7. Combinatorial properties of $\mathbf{t}_{3/2}$ and $\mathbf{t}'$
  8. On the sequence of first differences
  9. From the sequence $\Delta(\mathbf{t}_{3/2})$ back to the original sequence $\mathbf{t}_{3/2}$
  10. Uniform recurrence.
  11. Occurences at odd and even positions.
  12. Bit-wise complement.
  13. Reversal.
  14. Frequencies of letters in $\mathbf{t}_{3/2}$
  15. Some abstract harmonic analysis
  16. ...and 7 more sections

Key Result

Lemma 4

Let $\varphi:{\tt 0}\mapsto {\tt 010}$ and ${\tt 1}\mapsto {\tt 101}$. We have $\mathbf{t}'=\varphi(\mathbf{t}_{3/2})$.

Figures (7)

  • Figure 1: Estimation of the frequency of 0 in prefixes of the Thue--Morse word $\mathbf{t}_{3/2}$ in base $3/2$: on the left, for prefixes with length $\le 2000$; on the right, those with length in $[14000,15000]$.
  • Figure 2: The first levels of the tree associated with expansions in base $3/2$.
  • Figure 3: A DFAO generating the Thue--Morse sequence $\mathbf{t}_{3/2}$ in base $3/2$.
  • Figure 4: A DFAO generating the sequence of first differences $\Delta(\mathbf{t}_{3/2})$ of the Thue--Morse sequence $\mathbf{t}_{3/2}$ in base $3/2$.
  • Figure 5: Using the fixed point structure, we iterate the map $\Phi$ three times on successive prefixes of $\mathbf{t}_{3/2}$.
  • ...and 2 more figures

Theorems & Definitions (41)

  • Definition 1
  • Definition 2
  • Definition 3: Our sequence of interest
  • Lemma 4
  • proof
  • Definition 5
  • Example 6
  • Theorem 7: Cassaigne-Karhumaki-1997
  • Proposition 8: Cassaigne-Karhumaki-1997
  • Lemma 9
  • ...and 31 more