Diophantine Approximation, Khintchine's Theorem, Torus Geometry and Hausdorff Dimension

A general form of the Borel-Cantelli Lemma and its connection with the proof of Khintchine's Theorem on Diophantine approximation and the more general Khintchine-Groshev theorem are discussed. The torus geometry in the planar case allows a relatively direct proof of the planar Groshev theorem for the set of ψ-approximable points in the plane. The construction and use of Haudsorff measure and dimension are explained and the notion of ubiquity, which is effective in estimating the lower bound of the Hausdorff dimension for quite general lim sup sets, is described. An application is made to obtain the Hausdorff dimension of the set of ψ-approximable points in the plane when ψ(q) = q −v , v > 0, corresponding to the planar Jarník-Besicovitch theorem. 1. DIOPHANTINE APPROXIMATION Diophantine approximation is a quantitative analyis of the density of the rationals in the reals. It is easy to see from the distribution of the integers Z in the real line R, that given any α ∈ R and any q ∈ N, there exists a p = p(α, q) ∈ Z such that |qα − p| 1/2 or |α − p/q| 1 2q. It is possible to do better using continued fractions (see [6, 18]) and thanks to Dirichlet's box argument [19], to obtain a best possible result. Theorem 1 (Dirichlet). Given α ∈ R, N ∈ N, there exist integers p, q with 1 q N such that |α − p/q| 1 q(N + 1). Given any ξ ∈ R, the convenient notation ξ := min{|ξ − p| : p ∈ Z} will be used. Thus Dirichlet's theorem implies that given any α ∈ R, there are infinitely many q ∈ N such that qα = min{|qα − p| : p ∈ Z} < 1 q. More generally, an error term or approximation function ψ : N → (0, ∞), where lim q→∞ ψ(q) = 0, is introduced and the solubility of qα < ψ(q) (1) considered (ψ(q) = 1/q in Dirichlet's theorem). Note that although restricting the approximation to ratio-nals p/q with (p, q) = 1 is natural and indeed is associated with the Duffin-Schaeffer conjecture (see [13]), coprimality does not arise in the present formulation. The point α is said to be ψ-approximable if (1) holds for infinitely many q ∈ N. The set W (ψ) of ψ-approximable numbers is invariant under translation by integers and so there is no loss …