2006.08563
Counting Points of Bounded Height in Monoid Orbits
Wade Hindes, Umberto Zannier
correctmedium confidence
- Category
- math.DS
- Journal tier
- Specialist/Solid
- Processed
- Sep 28, 2025, 12:55 AM
- arXiv Links
- Abstract ↗PDF ↗
Audit review
The paper’s Theorem 1.1 proves two-sided bounds of the form (log B)^b and (log B)^{b+ε} for #{f in MS : H(f(P)) ≤ B} under freeness, using Tate’s telescoping height lemma and a generating-function approach after rationally approximating log degrees . The candidate solution proves the same type of bounds by a different route: a renewal-type recursion for degree products and a Dirichlet-series threshold at δ defined by ∑ d_i^{-t} = 1, combined with height control via h(φ(Q)) = d h(Q) + O(1) . Both arguments yield exponents arbitrarily close to the same critical δ (as illustrated numerically in the paper’s example) .
Referee report (LaTeX)
\textbf{Recommendation:} minor revisions
\textbf{Journal Tier:} specialist/solid
\textbf{Justification:}
The work gives a clean and effective framework for height-counting in monoid orbits, sharpening to matching upper/lower (log B)-power laws in the free-monoid case. The methodology—transferring from heights to degrees via Tate’s telescoping lemma, then counting via generating functions—is well-selected and robust. Clarity could be improved in a few technical steps (e.g., explicit statement of the critical exponent and discussion of optimal constants), but overall the results are sound and of solid interest to arithmetic dynamics.