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The following problem is inspired by a problem in Pugh's Mathematical Analysis book. (Chapter 2 Problem 42).

In the problem he asks one to consider the standard Cantor set on the unit interval, and then to map it to the unit circle using the standard mapping $z \mapsto e^{2 \pi i z}$. Then given this set $D \subset S^1$ one can consider the subset of chords between all points in $D$. Namely $Chords(D) = \{t x + (1-t)y ~|~ x, y \in D, t \in [0,1]\}$. He asks if this set is compact and/or convex, to which the answers are yes and no respectively.

A further question can be asked: Does this set have measure zero? I believe naive bounds would show that if one constructs a cantor set by removing over $1/2$ of the interval at each iteration we indeed have the chords have measure 0. I am unsure of the result in the standard case of removing middle thirds.

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I agree with questioner that the area measure is positive for a nice cantor set if and only if the hausdorf dimension of the cantor set is at least 1/2.There are nice examples of such in the theory of convex cocompact Fuchsian groups. The set in question is diffeomorphic to the set of full non euclidean geodesics contained in the convex hull of the limit set of the group and has been studied a lot. (Dennis Sullivan)

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