A zero-one law for improvements to Dirichlet’s Theorem

Abstract

We give an integrability condition on a function ψ \psi guaranteeing that for almost all (or almost no) x ∈ R x\in \mathbb {R} , the system | q x − p | > ψ ( t ) |qx-p|> \psi (t) , | q | > t |q|>t is solvable in p ∈ Z p\in \mathbb {Z} , q ∈ Z ∖ { 0 } q\in \mathbb {Z}\smallsetminus \{0\} for sufficiently large t t . Along the way, we characterize such x 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.