arXiv:2310.00122v1
Dimension bounds for escape on average in homogeneous spaces
math.DS37A1737A25
Abstract
Let , where is a Lie group and is a uniform lattice in , and let be an open subset of . We give an upper estimate for the Hausdorff dimension of the set of points whose trajectories escape on average with frequency , where .
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Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.
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Detailed mathematical audit
01Statements2 reported findingsCorrect
The Hausdorff-codimension estimates for trajectories that escape an open set on average are correct for compact homogeneous spaces with exponentially mixing diagonalizable flow; one domain description needs a minor formal correction.
Theorems 1.2 and 1.5Correct
The global and horospherical codimension bounds are correct
Pages 3–6 and 8–18 · Theorems 1.2 and 1.5 · arXiv:2310.00122v1
Effective equidistribution bounds the proportion of each Bowen block with an excessive complement average. The entropy-like function records the combinatorial choice of bad subblocks, and the covering exponent gives the displayed codimension on an unstable subgroup and then on the full space.
Full paper, version 1 ↗Definition of the auxiliary functionMinor formal correction · no status impact
Continuity is only asserted on the displayed square
Page 3 · equations (1.4)–(1.5) · arXiv:2310.00122v1
The function is defined on the unit square, where the logarithms and endpoint convention make it continuous, but the next sentence says it is continuous on the whole real plane. Replacing real plane by unit square is the unique correction and does not alter any theorem.
02Proofs2 reported findingsCorrect
The effective-equidistribution covering argument and the unstable-to-global product reduction are correct and complete after the domain correction.
Proof of Theorem 1.5Correct and complete
The Bowen-box cover has the claimed exponent
Pages 8–16 · proof of Theorem 1.5 · arXiv:2310.00122v1
Smoothing the inner core gives uniform equidistribution error, a binomial decomposition records blocks with excessive escape, and tessellation of the unstable subgroup converts the count into a Hausdorff cover. Optimization in the block length yields the stated auxiliary function.
Deduction of Theorem 1.2Correct and complete
Stable and neutral directions restore the full-space estimate
Pages 6–8 · proof of Theorem 1.2 · arXiv:2310.00122v1
The local multiplication map splits the group into unstable and nonexpanding factors. Thickening an escaping point in the nonexpanding direction changes membership only by the chosen inner-core radius, so the slice codimension transfers unchanged to local charts and hence globally.
03Novelty0 reported findingsNo non-novelty findings
No non-novelty findings.