Diophantine Approximation, Flows on Homogeneous Spaces and Counting

Let ξ ∈ R, let ι : [1,∞)→ (0, 1] be a positive decreasing function, and let E ξ (ι, Q) be the set of integer pairs (p, q) satisfying |p + qξ| < ι(q), 0 < q < Q. In a series of papers starting in 1959 Erdős [14], Schmidt [22, 23], Lang [18, 19, 9], Adams [1, 2, 3, 4, 5, 7, 6, 8], Sweet [26], and others, considered the problem of finding the asymptotics for the cardinality |E ξ (ι, Q)| as Q gets large. Schmidt [22] has shown that for almost every ξ ∈ R the asymptotics are given by the volume of the corresponding subset of R, provided the latter tends to infinity. Precursors of this important result are due to LeVeque [21] and Erdős [14]. Lang [18] proved |E ξ (1/q,Q)| ∼ cξ log(Q) for quadratic ξ. In [19] he also proved a result for a general class of ξ but with stronger constraints on the rate of decay of ι. Adams [1, 2] established various counting results for more specific ξ. Finally, Schmidt [23], Adams [23, 3, 7, 6, 8] and Sweet [26] generalized some of these results to simultaneous approximations. Opposed to the above “localised” setting, where the bound on |p+qξ| is expressed as a function of q, we consider the “non-localised” (sometimes called “uniform”) situation, where the bound is expressed as a function of Q. Let ξ be a vector in R, and, throughout this article, let ψ : [1,∞) → [1,∞) be a function. We consider the set