Recurrent dimensions of quasi-periodic solutions for nonlinear evolution equations. II: Gaps of dimensions and Diophantine conditions (Q1884229)

[For part I see: Trans. Am. Math. Soc. 354, 1137--1151 (2002; Zbl 0986.35011)]. This paper continues the author's exploration of the relationship between the fractal geometry of quasi-periodic orbits and delicate Diophantine properties of the frequencies involved. Here, the case of two frequencies and irrational frequencies of the Khinchine-Lévi class is studied. Particular attention is paid to the gap between the upper and lower recurrent dimensions, which is put forward as a central parameter reflecting the chaotic behaviour of the orbits.