A zero-one Law for improvements to Dirichlet's Theorem

Abstract

We give an integrability condition on a function ψ guaranteeing that for almost all (or almost no) x ∈ R, the system |qx − p| < ψ(t), |q| < t is solvable in p ∈ Z, q ∈ Z {0} for sufficiently large t. Along the way, we characterize such x in terms of the growth of their continued fraction entries, and we establish that Dirichlet's Approximation Theorem is sharp in a very strong sense. Higher-dimensional generalizations are discussed at the end of the paper.

Hausdorff Measure of Sets of Dirichlet Non‐improvable Numbers

“…Moreover, as noticed by Kleinbock & Wadleigh [10], D(ψ) c = ∅ whenever ψ is non-increasing and tψ(t) < 1 for all t ≫ 1.…”

“…Our approach to the problems discussed in this paper is reasonably general. Although, together with the results of [10], we have almost complete information on the size of sets of Dirichlet non-improvable real numbers in the one-dimensional setting, there are still some open problems which could improve the current state of knowledge. We list some of them here.…”

“…what is the size of the complement D(ψ) c , in the sense of measure/dimension, for functions ψ which decay slower than (1 − ǫ)ψ 1 for any ǫ > 0? Beyond some particular choices of ψ, nothing was known until recently, when Kleinbock & Wadleigh [10] proved a dichotomy law on the Lebesgue measure of D(ψ).…”

“…Continued fractions and improving Dirichlet's theorem. The starting point for the work of Davenport & Schmidt [4] and Kleinbock & Wadleigh [10] is an observation that Dirichlet improvability is equivalent to a condition on the growth rate of partial quotients. Let's recall this connection.…”

“…We also write p n /q n = [a 1 , ..., a n ] (p n , q n coprime) for the n'th convergent of x. The results of [4,10] rely crucially on the following observations, proved in Lemma 2.5 below:…”