2108.08567
ON THE DENSENESS OF SOME SPARSE HOROCYCLES
Cheng Zheng
correcthigh confidence
- Category
- math.DS
- Journal tier
- Specialist/Solid
- Processed
- Sep 28, 2025, 12:56 AM
- arXiv Links
- Abstract ↗PDF ↗
Audit review
The paper explicitly proves both density statements for every nonperiodic p on any nonuniform Γ ≤ PSL(2,R): Theorem 1.1 establishes density of {u(n^{1+γ})p} for all sufficiently small γ>0, and Theorem 1.2 asserts density along N-almost-prime times for some fixed N; both are stated and proved in full generality with clear proof sketches and detailed lemmas (e.g., reduction to U, periodic approximation, Fourier bounds, and a Jurkat–Richert linear sieve coupled with effective discrete horocycle equidistribution) . The candidate solution asserts the problems remained open as of 2021; this is contradicted by the paper’s results and proofs.
Referee report (LaTeX)
\textbf{Recommendation:} no revision
\textbf{Journal Tier:} specialist/solid
\textbf{Justification:}
This concise note establishes topological density along two natural sparse subsets of times (slightly superlinear polynomial times and almost-prime times) for horocycle flows on any finite-area hyperbolic surface, resolving questions previously open in this full generality. The proofs are clean, rely on standard effective tools, and are presented clearly enough for expert readers. No issues affecting correctness were found.