arXiv:1701.04139v2
An application of lattice points counting to shrinking target problems
math.DS37D4053D2537A25
Abstract
We apply lattice points counting results of Gorodnik and Nevo to solve a shrinking target problem in the setting of geodesic flows on hyperbolic manifolds of finite volume.
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Detailed mathematical audit
01Statements2 reported findingsCorrect
The convergence and divergence shrinking-target laws for discrete geodesic flow on finite-volume hyperbolic manifolds are correct under the stated monotonicity and rate conditions.
Theorems 1.1 and 1.2Correct
The zero-one shrinking-target laws are correct
Pages 2–4 and 11–16 · Theorems 1.1–1.2 · arXiv:1701.04139v2
Convergence of the target-volume sum gives the finite-hit conclusion directly by Borel-Cantelli. In the divergent case, the stated lower-rate condition makes the lattice-counting error negligible relative to the main term, yielding the required quasi-independence.
Full paper, version 2 ↗Theorem 1.3Correct
The block-sum generalization has the correct quantitative hypothesis
Pages 4–5 and 14–16 · Theorem 1.3 · arXiv:1701.04139v2
The theorem isolates exactly the block lower bound needed for the second-moment argument. The earlier radius condition is then verified as a sufficient special case, rather than being silently used in the general statement.
02Proofs2 reported findingsCorrect
The lattice-point estimate and dynamical Borel-Cantelli argument are correct and complete.
Theorem 2.2Correct and complete
The abstract second-moment criterion is applied with the right error term
Pages 5–7 · Theorem 2.2 · arXiv:1701.04139v2
Pair intersections are compared with products of target measures plus the counting error, and the resulting variance is little-o of the square of the cumulative expectation on the stated blocks. This gives infinitely many hits almost surely.
Sections 3–4Correct and complete
Hyperbolic lattice counting supplies every geometric estimate
Pages 7–16 · counting theorems and completion · arXiv:1701.04139v2
The fundamental-domain decomposition treats compact and cuspidal pieces uniformly, the main term matches hyperbolic ball volume, and the exponential error is inserted with the correct radius dependence. The final summation covers both the concrete and block formulations.
03Novelty0 reported findingsNo non-novelty findings
No non-novelty findings.