[Paper Review] A zero-one law for uniform Diophantine approximation in Euclidean norm

Dmitry Kleinbock, Anurag Rao|arXiv (Cornell University)|Sep 30, 2019

Mathematical Dynamics and Fractals24 references4 citations

TL;DR

This paper establishes a zero-one law for uniform Diophantine approximation in the Euclidean norm by analyzing shrinking target problems for diagonal flows on the space of lattices. Using tools from the geometry of numbers and dynamical systems, it proves that for a given approximation function ψ, the set of real numbers α satisfying a specific norm-based inequality has either full or zero Lebesgue measure, depending on the convergence or divergence of a related series, generalizing prior results in the supremum norm case.

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 $\mathbb{R}^2$ 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.

Motivation & Objective

• To extend the zero-one law for uniform Diophantine approximation from the supremum norm to arbitrary norms, particularly the Euclidean norm.
• To analyze shrinking target problems for one-parameter diagonal flows on the space of lattices in R².
• To characterize the measure-theoretic behavior of sets of real numbers satisfying a norm-sensitive approximation inequality.
• To generalize results of Andersen and Duke on Minkowski's theorem improvement and measure-zero sets in Diophantine approximation.
• To establish a dynamical Borel–Cantelli lemma framework for approximation in the Euclidean norm setting.

Proposed method

• Model the Diophantine approximation problem as a shrinking target problem for diagonal flows on the space of unimodular lattices in R².
• Use the action of the unipotent flow u_α on lattices to translate approximation conditions into lattice vector inclusion in scaled norm balls.
• Apply the dynamical Borel–Cantelli lemma of Maucourant to derive a zero-one law for the set of α satisfying the approximation inequality.
• Employ geometry of numbers techniques to analyze the critical locus of the norm ball and verify transversality to the parabolic subgroup's Lie algebra.
• Prove transversality of the critical locus to the distribution generated by the Lie algebra of upper-triangular matrices via differential analysis of a local parametrization of the boundary of the norm ball.
• Use Mahler’s theorem to reduce the reducible case to the irreducible one by embedding a critical irreducible body B' into a general convex body B with equal critical determinant.

Experimental results

Research questions

• RQ1 Does a zero-one law hold for uniform Diophantine approximation in the Euclidean norm, analogous to known results in the supremum norm?
• RQ2 What is the measure-theoretic behavior (full or zero Lebesgue measure) of the set of real numbers α for which the inequality (αq−p)²/ψ(t)² + (q/t)² < 2/√3 has non-trivial integer solutions for all large t?
• RQ3 How does the geometry of the norm ball, particularly its critical locus, influence the dynamics of diagonal flows on the space of lattices?
• RQ4 Can the transversality condition required for the dynamical Borel–Cantelli lemma be established for general norms, especially the Euclidean norm?
• RQ5 To what extent does the critical determinant of a convex body determine the structure of its critical lattice locus?

Key findings

• A zero-one law holds for the set of α ∈ ℝ such that the inequality (αq−p)²/ψ(t)² + (q/t)² < 2/√3 has non-trivial integer solutions for all large t, depending on the convergence or divergence of the series ∑ −log(1−kψ(k))(1/k − ψ(k)).
• The set of α satisfying the approximation condition has full Lebesgue measure if the series diverges, and measure zero if it converges.
• The critical locus of the Euclidean norm ball is transversal to the distribution generated by the Lie algebra of upper-triangular matrices, a key condition for applying the dynamical Borel–Cantelli lemma.
• The transversality condition is established via differential analysis of a local parametrization of the boundary of the norm ball, showing that the bottom-left entry of the derivative matrix is non-zero.
• For reducible convex bodies, the critical locus is contained in that of an irreducible body with the same critical determinant, allowing reduction to the irreducible case.
• The result generalizes earlier work of Andersen and Duke on measure-zero and uncountable sets of numbers for which Minkowski's theorem can be improved, now in the context of the Euclidean norm.