Hausdorff dimension of differences of badly approximable sets

Dorsa Hatefi, David Simmons

correcthigh confidence

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

Audit review

The uploaded paper proves exactly the statement in question: for every non-integer shift γ, the difference Bad^γ \ Bad has full Hausdorff dimension (stated as Theorem 1.1) using a new variant of Schmidt’s game (the rapid game) and a dynamical reformulation on unimodular grids; rapid-winning implies full Hausdorff dimension and Bad^γ \ Bad is shown to be rapid-winning, yielding the result . The candidate solution does not supply an independent proof; it cites and summarizes the same 2024 result and its method (rapid game on grids), so it aligns with the paper’s approach. Minor editorial note: the paper briefly mentions “Bad^γ ∩ Bad is strong winning” as motivational context but then proves the needed default strategy directly, so no gap arises from that remark .

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} specialist/solid

\textbf{Justification:}

The paper establishes a natural and interesting full-dimension result for inhomogeneous Diophantine approximation using a clean and arguably versatile game-theoretic tool (the rapid game). The argument builds on standard dynamical correspondences and contributes a new variant of Schmidt’s game tailored to the setting. Exposition is generally clear, and the methods appear correct; a few clarifications would further strengthen the presentation.