%0 Journal Article
%A Dodson, M. M.
%A Poschel, J.
%A Rynne, B. P.
%A Vickers, J. A. G.
%T The Hausdorff Dimension of Small Divisors for Lower-Dimensional KAM-Tori
%D 1992
%R 10.1098/rspa.1992.0155
%J Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences
%P 359-371
%V 439
%N 1906
%X Given a bounded domain [Note: See the image of page 359 for this formatted text] $\Omega \subset R^{m}$ and a Lipschitz map [Note: See the image of page 359 for this formatted text] $\phi $: $\Omega \mapsto R^{n}$, we determine the Hausdorff dimension of sets of points $\omega \in \Omega $ for which the inequality $|$k$\cdot \omega $-l$\cdot \phi $($\omega $)$|$ < $\psi $($|$k$|$+$|$l$|$) has infinitely many distinct integer solutions [Note: See the image of page 359 for this formatted text] (k, l) $\in Z^{m}\times Z^{n}$ satisfying $|$l$|\leq $ h, where h is a fixed integer. These sets `interpolate' between the cases h = 0 and h = $\infty $, 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 $\psi $ at infinity. Part of this result even extends to the case n = $\infty $ of infinite co-dimension, which is relevant in the KAM-theory of certain nonlinear partial differential equations.%U http://rspa.royalsocietypublishing.org/content/royprsa/439/1906/359.full.pdf