I'm trying to read this paper: Bourgain, J.; Jitomirskaya, S., Continuity of the Lyapunov exponent for quasiperiodic operators with analytic potential, J. Stat. Phys. 108, No. 5-6, 1203-1218 (2002), arXiv:math-ph/0110040. Zbl 1039.81019. CiteSeerX

But I don't understand the norm they are using in (1.3). They write down the condition $\vert\vert k\omega\vert\vert>C(\vert k\vert \log(1+\vert k\vert)^A)^{-1}$

($\omega$ is an irrational number, and I think $k$ is an integer), and say that this expression is a "strong Diophantine condition" on $\omega$.

Another line in the paper which may be illuminating is

"for $\vert k\vert\leq q/2, \vert kw-ka/q\vert<1/2q$ and hence $\vert\vert k\omega\vert\vert >1/2q$."

Here $a/q$ is a lowest terms fraction with property $\vert \omega-a/q\vert<1/q^2$.

Context suggests to me that the norm should be a related to how easy it is to approximate an irrational number with rationals, but they don't have a definition in the paper, and I don't think I understand Diophantine approximation well enough to guess what it is.