arXiv:math/0506514v1
Measure rigidity and -adic Littlewood-type problems
math.NTmath.DS11J6137A13
Abstract
The paper investigates various -adic versions of Littlewood's conjecture, generalizing a set-up considered recently by de Mathan and Teulie. In many cases it is shown that the sets of exceptions to these conjectures have Hausdorff dimension zero. The proof follows the measure ridigity approach of Einsiedler, Katok and Lindenstrauss.
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Detailed mathematical audit
01Statements2 reported findingsCorrect
The measure-rigidity consequences for -adic Littlewood-type problems and the recurrence alternatives for the associated diagonal actions are correct.
Theorems 1.1–1.3Correct
Measure rigidity and -adic Littlewood consequences
Pages 2–5 · Theorems 1.1–1.3 · arXiv:math/0506514v1
A counterexample yields a bounded orbit for the higher-rank diagonal action. Positive entropy for a nontrivial element invokes the stated measure-classification theorem, while zero entropy and recurrence force the symbolic and continued-fraction restrictions claimed. The alternatives cover every invariant limit measure obtained from the orbit averages.
Full paper, version 1 ↗Propositions in Sections 4–6Correct
Dynamical reformulation and recurrence criteria
Sections 4–6 · arXiv:math/0506514v1
The lattice embedding represents the real and -adic approximation factors by the corresponding short-vector product. Mahler compactness turns a positive lower bound into boundedness, and the recurrence argument preserves the required coordinate restrictions.
02Proofs2 reported findingsCorrect
The homogeneous-dynamics reduction and measure-rigidity argument are correct. One sentence reverses ‘bounded’ and ‘unbounded’; the surrounding equivalence and all later uses determine the harmless correction.
Sections 2–6Correct and complete
Invariant measures yield the announced alternatives
Sections 2–6 · arXiv:math/0506514v1
Orbit averages have invariant weak limits because the assumed counterexample keeps the orbit in a compact set. Entropy and measure classification control the positive-entropy case; the zero-entropy recurrence argument gives the complementary complexity restriction. Each conclusion is translated back through the same lattice embedding.
Proof of Theorem 1.2Typo
The orbit condition is called unbounded instead of bounded
Page 13 · proof of Theorem 1.2 · arXiv:math/0506514v1
After defining the exceptional set by a positive liminf, the proof says membership is equivalent to the orbit being ‘unbounded, i.e.’ contained in a fixed compact set . Replace ‘unbounded’ by ‘bounded’. Containment in is the definition used immediately afterward and is exactly what the positive liminf gives.
03Novelty0 reported findingsNo non-novelty findings
No non-novelty findings.