2410.19640
On the dimension of αβ-sets
Michael Hochman
correctmedium confidence
- Category
- Not specified
- Journal tier
- Strong Field
- Processed
- Sep 28, 2025, 12:56 AM
- arXiv Links
- Abstract ↗PDF ↗
Audit review
The paper proves the existence of an αβ-set of box dimension 1/2 via a refined Katznelson-style concatenation/perturbation scheme (Theorem 1.1), building two closed words Un,Vn at each stage, enforcing a growth condition, and passing to a limit set E that is the closure of an infinite αβ-orbit; the construction yields separated sets and covering estimates at natural scales εn and concludes dimB E = 1/2 (see Section 3 and Lemmas 3.3–3.5, along with the parameter choice leading to (12) and the dimension computation) . The candidate solution mirrors this scheme closely (two-word concatenation, fast growth, perturbative closing, and a standard diagonal avoidance to secure Q-independence of 1,α,β), differing mainly in notation and minor organizational details.
Referee report (LaTeX)
\textbf{Recommendation:} minor revisions
\textbf{Journal Tier:} strong field
\textbf{Justification:}
The paper settles a sharpness question about αβ-sets by constructing an example with box dimension 1/2, matching the Feng–Xiong lower bound. The construction is clear, quantitative, and carefully analyzed; additional sections on density-thinning and Assouad dimension broaden the contribution. Two minor points merit revision: make explicit the standard diagonal avoidance to ensure 1,α,β are Q-independent in the main construction and state the lower bound half of the box-dimension calculation alongside the upper bound.