This is related to this other question of mine about a paper of Colin and Honda.

I'm trying to follow the proofs line by line. I found the following piece of notation that is not explained in the text so i ask for its meaning or a reference where it is used other than this. I quote the beginning of page 6 in the cited paper

Since $d(h(\gamma_0), \gamma_0) = N$, it follows that $d(h(\alpha), h(\gamma_0)) \approx N$ for every $\alpha$. (Here $\approx$ means approximately equal to.)

For some context: $\gamma_0, \alpha$ are simple closed curves in a surface, $h$ is an automorphism of the surface, $d(\cdot, \cdot)$ is the distance in the complex of curves (which is a natural number) and $N$ is a natural number. Hence, we are saying that two natural numbers are approximately equal?

Since $N$ is a number that, in that context, is arbitrarily large one could think that it means that $\lim_{N\to \infty} d(h(\gamma_0), \gamma_0) - N = 0$? But later on the text a similar exppresion appears changing $N$ by $0$ (end of page $7$).

So I would appreciate someone explaining to me what it means.

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    $\begingroup$ You didn't say that $d(h(\alpha), h(\gamma_0))$ is a natural number. Could it be that $d(h(\alpha), h(\gamma_0))$ is a real number, close to $N$ in the sense of distance ${} < 1/2$ ? $\endgroup$ – Gerald Edgar Jun 8 '19 at 0:21
  • $\begingroup$ sorry, I forgot to say that the distance in the complex of curves is a natural number as well. (it is explained in page 3 of the paper) $\endgroup$ – Paul Jun 8 '19 at 0:23
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    $\begingroup$ They just mean that the numbers are a uniformly bounded distance from N. Here the $\alpha$ are distance one from $\gamma_0$. $\endgroup$ – Autumn Kent Jun 8 '19 at 1:11
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    $\begingroup$ Uniform in $\alpha$, I think. That is, two functions of $\alpha$ are approximately equal if their difference is a bounded function of $\alpha$. At any rate, uniform in whatever is varying. $\endgroup$ – Noah Snyder Jun 8 '19 at 7:45
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    $\begingroup$ "Approximately equal" is never a phrase that would be used for individual specific natural numbers, it's always going to be used in reference to some kind of function. In this case one of the functions is constant, but if they're both constants then they'd be approximately equal no matter how far apart those constants are. $\endgroup$ – Noah Snyder Jun 8 '19 at 16:47

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