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I'm trying to read this paper:


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.

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For people who don't want to download the paper, or don't have a subscription to the journal, please put some actual detail in this question. e.g. the formula for the norm, and write out the context yourself. Read for reasons why. – David Roberts Jan 23 '12 at 22:42
Is it just the distance to the integers? – Charles Matthews Jan 23 '12 at 23:11
Charles: I don't think so. Or at least that suggestion doesn't clarify the second example in particular. My guess is that the norm is related to c(x) in the wikipedia page – Darren Ong Jan 23 '12 at 23:40
Darren, I don't have access to Springer, but already your first borrowing makes clear that $k$ means and integer and $\|\ \|$ is the distance from a real number to the nearest integer (without the "strong" appearance of $\log(1+|k|)$, the estimate is guaranteed by Dirichlet's theorem. – Wadim Zudilin Jan 24 '12 at 0:32
Wadim and Charles, you're right. After thinking through more clearly, it has to be the distance to the integers. Thanks a bunch! – Darren Ong Jan 24 '12 at 0:35

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