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Escape of mass of the Thue-Morse sequence

Erez Nesharim, Uri Shapira, Noy Soffer Aranov

TL;DR

The paper studies escape of mass for measures arising from quadratic Laurent series over $F_q((t^{-1}))$, focusing on the Thue–Morse sequence in $F_2((t^{-1}))$. It constructs diagonally-aligned number walls $ rak F(oldsymbol{ au})$ and proves they are explicitly generated by a 2D substitution tiling via a 4-coding, yielding a precise combinatorial handle on zeros in Hankel/Toeplitz determinants encoding Diophantine approximations. The main finding is that for the Thue–Morse sequence $oldsymbol{ au}$, the mass escape along left-shifts $t^joldsymbol{ au}$ is not full: $ lim_{d o ty} liminf_{j o ty} lim_{k o ty} e_{t^joldsymbol{ au}}(d,k)= frac{2}{3}$. Moreover, the authors exhibit subsequences along which full escape occurs, showing a nuanced, structured escape phenomenon and highlighting rich symmetries in the Thue–Morse wall. These results provide a concrete counterexample to the full-escape conjecture and reveal deep connections between automatic tilings, Diophantine approximation in function fields, and homogeneous dynamics.

Abstract

Every Laurent series in $\mathbb{F}_q\left(\left(t^{-1}\right)\right)$ has a continued fraction expansion whose partial quotients are polynomials. De Mathan and Teulié proved that the degrees of the partial quotients of the left-shifts of every quadratic Laurent series are unbounded. Shapira and Paulin and Kemarsky improved this by showing that certain sequences of probability measures on the space of lattices in the plane $\mathbb{F}_q\left(\left(t^{-1}\right)\right)^2$ exhibit positive escape of mass and conjectured that this escape is full -- that is, that these probability measures converge to zero. We disprove this conjecture by analysing in detail the case of the Laurent series over $\mathbb{F}_2$ whose sequence of coefficients is the Thue-Morse sequence. The proof relies on the discovery of explicit symmetries in its number wall.

Escape of mass of the Thue-Morse sequence

TL;DR

The paper studies escape of mass for measures arising from quadratic Laurent series over , focusing on the Thue–Morse sequence in . It constructs diagonally-aligned number walls and proves they are explicitly generated by a 2D substitution tiling via a 4-coding, yielding a precise combinatorial handle on zeros in Hankel/Toeplitz determinants encoding Diophantine approximations. The main finding is that for the Thue–Morse sequence , the mass escape along left-shifts is not full: . Moreover, the authors exhibit subsequences along which full escape occurs, showing a nuanced, structured escape phenomenon and highlighting rich symmetries in the Thue–Morse wall. These results provide a concrete counterexample to the full-escape conjecture and reveal deep connections between automatic tilings, Diophantine approximation in function fields, and homogeneous dynamics.

Abstract

Every Laurent series in has a continued fraction expansion whose partial quotients are polynomials. De Mathan and Teulié proved that the degrees of the partial quotients of the left-shifts of every quadratic Laurent series are unbounded. Shapira and Paulin and Kemarsky improved this by showing that certain sequences of probability measures on the space of lattices in the plane exhibit positive escape of mass and conjectured that this escape is full -- that is, that these probability measures converge to zero. We disprove this conjecture by analysing in detail the case of the Laurent series over whose sequence of coefficients is the Thue-Morse sequence. The proof relies on the discovery of explicit symmetries in its number wall.

Paper Structure

This paper contains 21 sections, 25 theorems, 153 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Partial escape of mass
  3. Tilings and diagonally-aligned number walls
  4. The main theorem
  5. The basic tiles and their group of symmetries
  6. Actions on matrices
  7. The $2$-substitution
  8. The $4$-coding
  9. Structure of the paper
  10. The frame relations
  11. Number walls
  12. Diagonally-aligned number walls
  13. The tiling and its structure
  14. The structure of $\sigma^\infty(\mathbf{a}_{0})$
  15. Applying the coding
  16. ...and 6 more sections

Key Result

Theorem 1.2

Figures (4)

  • Figure 1: Hankel determinants of the Thue-Morse sequence $\mathfrak{H}_{m,n}(\alpha)$ for every $m,n\geq1$ such that $2m+n-2\leq128$, where the $0$'s are white and the $1$'s are black.
  • Figure 2: The (diagonally-aligned) number wall of the Thue-Morse sequence $\mathfrak{F}_{m,n}(\alpha)$ for $0\leq m\leq129$ and $1\leq n\leq130$, where the $0$'s are white and the $1$'s are black.
  • Figure 3: The (diagonally-aligned) number wall of the Thue-Morse sequence $\mathfrak{F}_{m,n}(\alpha)$ for $0\leq m\leq2049$ and $1\leq n\leq2050$, where the $0$'s are white and the $1$'s are black.
  • Figure 4: SageMath output of the pattern $\sigma^{5}(\mathbf{a}_{0})$, where the tile symbols have been altered in a suggestive manner.

Theorems & Definitions (65)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • Example 1.7
  • Definition 1.8
  • Lemma 1.9
  • proof
  • ...and 55 more