I'm slightly familiar with that paper of Bjorn's, and I don't think they address the limiting density question at all. (They do confirm that there are no multiple intersections when n is odd, which Dan mentioned parenthetically in his initial post.) But Gareth's hand-wavy proof seems quite right to me, and not even hard to shore up. Every set of four distinct points on the circle gives rise to exactly one intersection point in the interior of the circle (where ac crosses bd, if they are arranged in cyclic order abcd). The limit distribution that Dan asked about is just the uniform distribution on the 4-torus (minus the diagonal where two coords are equal) pushed forward to the disk. This is just another way to say what Gareth already pointed out: if you take any region of the disk and want to know its weight in this distribution, all you need to do is take its inverse image under that map and get its measure. This distribution will not only give you the limit of intersections of regular n-gons, but indeed will be the distribution of intersections for the chords of n randomly-chosen points on the circle as well. Oh -- but you should count distributions with multiplicity, when needed! If k lines all meet at a point, that's k-choose-2 intersections there. As that Poonen paper notes, there are points in the 30n-gon where seven (!) lines meet. If you perturb the points a little, you'll get 21 intersection points there; don't make your job harder by counting it only once. (Homework: what is the radius at which this biggest-ever intersection in the unit circle takes place?) --Michael Kleber On 2/2/06, dasimov@earthlink.net <dasimov@earthlink.net> wrote:

Gary writes:

<< I wrote:

<< Given integer n > 0, draw a line segment in C connecting each pair of nth roots of unity. Then as n -> oo, does the set of intersection points in C (assume each is given equal weight and the weights sum to 1) approach a continuous density on the unit disk? (Note: we care only about the intersection points, not the rest of the line segments.)

...

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There's a paper on this by Bjorn Poonen and Mike Rubinstein:

http://math.berkeley.edu/~poonen/papers/ngon.pdf

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That's a fantastic paper based on the theorem they announce; since Bjorn Poonen is one of only about 7 people who ever was in the Top Five in the Putnam Exam *four* times, I don't doubt their result is accurate.

But I glanced at each page, and searched for the words "density", "limit" and "limiting" but came up empty. Can anyone find where in the paper it addresses the question of a limiting density?

(Also, it appears that one of the references addresses which sets of nth roots of unity add up to 0, a question that arose in this venue not long ago.)

Thanks,

Dan

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