Escape of mass of the Thue-Morse sequence

2025-03-24

AI Summary

This paper investigates the continued fraction expansion of the Laurent series associated with the Thue–Morse sequence over the function field $mathbb{F}_2((t^{-1}))$, focusing on the unboundedness of degrees of partial quotient polynomials and the mass escape phenomenon of the associated lattice space measure. Employing a synthesis of function-field continued fraction theory, $p$-adic dynamical systems, and lattice space measure analysis—leveraging the algebraic-combinatorial structure of the Thue–Morse sequence—the authors refute the Shapira–Paulin–Kemarsky conjecture on full mass escape: they prove that the corresponding measure admits a nontrivial weak limit, implying incomplete mass escape. Furthermore, they discover and rigorously characterize a previously unknown explicit symmetric structure—the “number wall”—within the partial quotients, and establish a precise correspondence between this symmetry and the degree distribution of the partial quotients.

Technology Category

• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Machine Learning: Other Foundations of Machine Learning
• Game Theory and Economic Paradigms: Other Foundations of Game Theory & Economic Paradigms

Application Category

• Web Mining and Content Analysis: Bridging structured and unstructured data
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

Every Laurent series in $mathbb{F}_qleft(left(t^{-1} ight) ight)$ has a continued fraction expansion whose partial quotients are polynomials. De Mathan and Teuli'e 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 $mathbb{F}_qleft(left(t^{-1} ight) ight)^2$ 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.

Problem

• Disprove full escape of mass conjecture for Thue-Morse sequence
• Analyze Laurent series over F2 with Thue-Morse coefficients
• Investigate symmetries in number wall for continued fractions

Innovation

• Analyzed Thue-Morse sequence coefficients
• Discovered explicit number wall symmetries
• Disproved full escape of mass conjecture