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

2026-02-25

AI Summary

This study investigates the symbol frequencies of 0s and 1s in Thue–Morse-type binary sequences generated by integer expansions with base 3/2. By innovatively combining harmonic analysis on compact groups with substitution dynamical systems, the authors introduce linear operators on the 2-adic integers and employ spectral contraction arguments, further integrating techniques from Toeplitz sequence analysis and anti-substitution dynamics. This synthesis establishes a novel framework that proves the uniqueness of symbol frequencies, showing that both 0 and 1 occur with frequency exactly 1/2. The results confirm several conjectures posed by Dekking and reveal additional structural properties of the sequence’s factor set, including uniform recurrence, symmetry, and a rigidity manifested through dyadic lifts.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Cognitive Modeling & Cognitive Systems: Symbolic Representations
• Game Theory and Economic Paradigms: Other Foundations of Game Theory & Economic Paradigms

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Security and Privacy: Large-scale security measurements

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.

Problem

• Thue–Morse sequence
• symbol frequencies
• rational base numeration
• automatic sequences
• Dekking conjectures

Innovation

• harmonic analysis
• 2-adic integers
• Toeplitz sequence
• substitution dynamics
• symbol frequencies