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.
KAM Theory for Lower Dimensional Tori within the Reversible Context 2