A Zero-One Law for Uniform Diophantine Approximation in Euclidean Norm

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 1st-named author and Wadleigh [ 17], 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 the suitably scaled norm ball. We use methods from geometry of numbers to generalize a result due to Andersen and Duke [ 1] on measure zero and uncountability of the set of numbers (in some cases, matrices) 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 the dynamical Borel–Cantelli lemma of Maucourant [ 25] produces, given an approximation function $\psi $, a zero-one law for the set of $\alpha \in \mathbb{R}$ such that for all large enough $t$ the inequality $\left (\frac{\alpha q -p}{\psi (t)}\right )^2 + \left (\frac{q}{t}\right )^2 < \frac{2}{\sqrt{3}}$ has non-trivial integer solutions.

A note on Dirichlet spectrum

“…For n = 1 real number θ is singular if and only if it is rational. From another hand, for sup-norm |y y y| ∞ = max Davenport and Schmidt [9] as well as recent papers [17,18,19] with many metric results and the references therein…”

“…We will use natural coordinates and consider cylinders Π ′ ν+1 = GΠ ν+1 , Π ′ ν+1 = GΠ ν+1 . By Condition 3) according to (17) and by (57) we have…”

Abundance of Dirichlet-improvable pairs with respect to arbitrary norms

“…By Lemma 3.8, L(B) is a C 1 -submanifold of the space of all lattices in R 2 . See [KR,Corollary 7.5] for a proof of this. In particular we get a non-degenerate C 1 -parameterization…”

“…Then a standard argument usually referred to as the Dani correspondence gives the next proposition. It is a special case of a general correspondence described in [KR,Proposition 2.1]; we include the elementary proof for the sake of keeping the paper self-contained.…”

“…Fixing one critical lattice, this Lemma allows us to parameterize the critical locus of an irreducible B by traversing along the boundary ∂B from one lattice point to another (cf. [KR,Corollary 7.5]). We also use the result of Mahler that every convex symmetric domain contains an irreducible domain with the same critical determinant: C B ).…”