Some Remarks on the Notion of Bohr Chaos and Invariant Measures

Matan Tal

correctmedium confidence

Category: math.DS
Journal tier: Strong Field
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper defines Bohr chaoticity as a universal property: for every non-trivial bounded sequence a=(a_n) there must exist x and a continuous f with limsup_N (1/N)|∑_{n<N} a_n f(T^n x)|>0; see the definition in the Introduction . The paper proves this under specification by embedding a full shift via a unique-parsing construction (Theorems 4.1.1 and 4.2.1) , ensuring correlation with arbitrary non-trivial sequences. The candidate solution, however, constructs only a single arithmetic-progression-supported sequence a and shows positive correlation for that one, which does not satisfy the universal quantifier in the Bohr-chaos definition.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} strong field

\textbf{Justification:}

The work gives a clear, conceptually robust route from specification to Bohr chaoticity via unique parsing and a factor onto a full shift, tying Bohr chaos to classical dynamical notions. The results address questions raised in recent literature and integrate with known universality phenomena. Minor expansions would improve readability in a few sketched steps.