arXiv:1910.00126v5
A zero-one law for uniform Diophantine approximation in Euclidean norm
math.NT11J0411J1337A1737D40
Abstract
We study a norm sensitive Diophantine approximation problem arising from the work of Davenport and Schmidt on the improvement of Dirichlet's theorem. Its supremum norm case was recently considered by the first-named author and Wadleigh, and here we extend the set-up by replacing the supremum norm with an arbitrary norm. This gives rise to a class of shrinking target problems for one-parameter diagonal flows on the space of lattices, with the targets being neighborhoods of the critical locus of a suitably scaled norm ball. We use methods from geometry of numbers and dynamics to generalize a result due to Andersen and Duke on measure zero and uncountability of the set of numbers for which Minkowski approximation theorem can be improved. The choice of the Euclidean norm on corresponds to studying geodesics on a hyperbolic surface which visit a decreasing family of balls. An application of a dynamical Borel-Cantelli lemma of Maucourant produces a zero-one law for improvement of Dirichlet's theorem in Euclidean norm.
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Detailed mathematical audit
01Statements2 reported findingsCorrect
The absolute-winning theorem for norm-dependent Dirichlet-improvable numbers and the Euclidean critical and zero--one results are correct. Two local notation defects have unique corrections and do not change any theorem.
Theorems 1.3 and 3.1Correct
Norm-dependent Dirichlet-improvability has the stated winning and null properties
Pages 4 and 8--12 · Theorem 1.3 and the general measure-zero theorem · arXiv:1910.00126v5
The Dani correspondence identifies improvement with eventual avoidance of a neighborhood of the norm's critical locus. The critical-locus analysis supplies a uniformly avoidable piece for the absolute game, while ergodicity shows that a fixed strict improvement has measure zero. Taking the countable union over rational improvement constants preserves both the asserted null conclusion and absolute winning.
Theorems 1.4--1.6Correct
The Euclidean critical theorem and zero--one law follow from the lattice targets
Page 4 and Sections 4--6 (pages 13--22) · Euclidean main results · arXiv:1910.00126v5
The hexagonal critical locus gives . For a continuous non-increasing , the shrinking-target radius is monotone after the stated change of variables. Hyperbolic inner and outer ball estimates are both linear in near the critical radius, so Maucourant's integral test is equivalent to convergence or divergence of . The local product argument transfers the Haar zero--one statement to the horospherical parameter.
02Proofs3 reported findingsCorrect
The dynamical correspondence, critical-locus geometry, hyperbolic target comparison, and transverse zero--one argument are correct. The printed monotonicity word and one terminal radius are typographical errors with mechanically determined repairs.
Dani correspondence and critical locusCorrect and complete
The dynamical reformulation preserves all quantifiers
Sections 2--4 · Proposition 2.1 and Theorems 3.7 and 3.11 · arXiv:1910.00126v5
The diagonal rescaling has determinant one and converts the two approximation inequalities into avoidance of the compact target with radius . The critical-locus decomposition covers parallelogram and non-parallelogram norm balls, and the game strategy avoids the resulting smooth pieces without changing the improvement constant's strict inequality.
Introductory conventionTypo
The monotonicity direction for the approximation function is reversed in one sentence
Page 3 · paragraph immediately after the definition of norm-dependent approximation · arXiv:1910.00126v5
The sentence says that will be non-decreasing, whereas the definitions, Theorems 1.5--1.6, and every later use require to be non-increasing. Replacing `non-decreasing' by `non-increasing' is uniquely forced by the surrounding text and changes no argument.
Upper-half-plane target estimateTypo
The last lower bound has a missing square root
Page 15 · proof of Lemma 4.3 · arXiv:1910.00126v5
The proof obtains , hence . The final line prints . The target was defined using , so the square-root correction is mechanically determined and gives exactly the bound needed for the inclusion.
03Novelty0 reported findingsNo non-novelty findings
No non-novelty findings.