arXiv:1407.5310v2
Singular systems of linear forms and non-escape of mass in the space of lattices
math.DSmath.NT
Abstract
Singular systems of linear forms were introduced by Khintchine in the 1920s, and it was shown by Dani in the 1980s that they are in one-to-one correspondence with certain divergent orbits of one- parameter diagonal groups on the space of lattices. We give a (conjecturally sharp) upper bound on the Hausdorff dimension of singular systems of linear forms (equivalently the set of points with divergent trajectories) as well as the dimension of the set of points with trajectories 'escaping on average' (a notion weaker than divergence). This extends work by Cheung, as well as by Chevallier and Cheung, on the vector case. Our method differs considerably from that of Cheung and Chevallier, and is based on the method of integral inequalities developed by Eskin, Margulis and Mozes.
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Detailed mathematical audit
01Statements2 reported findingsCorrect
The Hausdorff-dimension upper bound for singular systems, the entropy estimate, and the non-escape-of-mass consequences are correct.
Theorem 1.1 and Corollary 1.2Correct
The escape-on-average dimension bound is correct
Pages 3–4 and 6–17 · Theorem 1.1 and Corollary 1.2 · arXiv:1407.5310v2
The covering estimate charges a fixed dimensional loss for each prescribed proportion of time outside a compact set. Under the Dani correspondence, singular matrices give complete escape on average, and insertion of the unstable-leaf dimension yields the displayed codimension.
Full paper, version 2 ↗Theorems 1.3 and 1.4Correct
The entropy and limiting-mass inequalities are correct
Pages 4–5 and 17–25 · Theorems 1.3–1.4 · arXiv:1407.5310v2
The compactly supported entropy estimate bounds the mass that can remain near the cusp, and upper semicontinuity on the compact part gives the stated lower bound on the mass of a weak limit. The normalization of entropy and the lost mass agree in the two statements.
02Proofs2 reported findingsCorrect
The combinatorial covering, height-function contraction, and entropy arguments are correct and complete.
Proof of Theorem 1.1Correct and complete
The multi-scale covers produce the claimed codimension
Pages 6–17 · proof of Theorem 1.1 · arXiv:1407.5310v2
Orbit segments are coded by their cusp visits, the number and diameter of covering boxes are bounded uniformly, and summing over visit patterns gives the announced Hausdorff exponent. The passage from finite blocks to limsup frequency is explicit.
Proofs of Theorems 1.3 and 1.4Correct and complete
Entropy controls escape of mass with the correct coefficient
Pages 17–25 · entropy and weak-limit proofs · arXiv:1407.5310v2
The height-function contraction creates a compact partition with controlled cusp atoms, and the entropy contribution of those atoms is bounded before taking limits. The ergodic decomposition and approximation steps preserve the coefficient appearing in the final mass inequality.
03Novelty0 reported findingsNo non-novelty findings
No non-novelty findings.