COUNTING AND HAUSDORFF MEASURES FOR INTEGERS AND p-ADIC INTEGERS

Davi Lima, Alex Zamudio Espinosa

correctmedium confidence

Category: Not specified
Journal tier: Specialist/Solid
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper proves D(Πp(F)) ≤ BD(F) (Theorem E) via a counting-measure contradiction argument and gives an explicit Z5 construction where D(Π5(F)) equals the upper box dimension and strictly exceeds the lower box dimension, with HD(F) no larger than the lower box dimension . The candidate solution independently proves the same inequality using a clean covering/interval argument and presents a different digit-restriction (Moran) construction in Z5 that also yields D(Π5(F)) = BDupper(F) > BDlower(F) and HD(F) = BDlower(F). The proofs align in substance on (a)–(b). For (c), the paper’s example uses a {0,2,4}-digit scheme with alternating “freeze/expand” blocks to separate upper and lower box dimensions, while the candidate uses long oscillating blocks with m_n ∈ {2,5}; both meet the existence claim. Minor notational inconsistencies in the paper (using BD for both upper and lower box dimensions) do not affect correctness but warrant revision .

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} specialist/solid

\textbf{Justification:}

The paper establishes a clear and correct connection between counting dimension on integers and box-counting dimension in p-adic integers, and it provides a concrete construction illustrating the possible gap between upper and lower box dimensions while matching the counting dimension. The main arguments are sound and well-motivated. Minor notational inconsistencies (upper vs lower box dimension) and a few terse steps in the example merit revision for clarity, but they do not affect correctness.