The Hausdorff dimension of small divisors for lower-dimensional KAM-tori

Given a bounded domain Ω ⊂ ℝ m and a Lipschitz map Φ : Ω ⟼ ℝ n , we determine the Hausdorff dimension of sets of points ω ∈ Ω for which the inequality | k ·ω — l·Φ(ω)| < Ψ (| k | + | l |) has infinitely many distinct integer solutions ( k, l )∈ℤ m x ℤ n satisfying | l | ⩽ h , where h is a fixed integer. These sets ‘interpolate’ between the cases h = 0 and h = ∞,which occur in the metric theory of Diophantine approximation of independent and dependent quantities, respectively. They arise, for example, in the perturbation theories of lower-dimensional tori in nearly integrable hamiltonian systems (KAM-theory). Among others, it turns out that their Hausdorff dimension is independent of h and n , it only depends on m and the lower order of Ψ at infinity. Part of this result even extends to the case n = ∞ of infinite co-dimension, which is relevant in the KAM-theory of certain nonlinear partial differential equations.