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If I remember correctly, there is a nice correspondence between continued fractions and convex hulls of lattice points in the plane. If $\theta>0$ is the slope of a line in $\mathbb{R}^2$ passing through the origin, let's take the convex hull of the lattice points ($(x,y)\in\mathbb{Z}^2$, $x,y\geq0$, $(x,y)\neq(0,0)$) lying on right of the line ($y\leq\theta x$); we get an infinite convex polygon. Let's do the same with the lattice points on the left, and let $(a_n,b_n)$ be the sequence of vertices of these polygons (ordered by increasing $a_n$). Is it true that $b_n/a_n$ is the $n$-th approximation of $\theta$ by its continued fraction? Can you point me to a nice proof of whatever is the correct form of this statement?

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Have you seen this? – J. M. Dec 7 '11 at 16:26
@J.M.: thanks a lot, the reference to Arnold's book is most useful! – user8268 Dec 7 '11 at 18:48

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