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A set of plane Measure 0 that contains a Circle of every Radius.
References
Falconer, K. J. The Geometry of Fractal Sets. New York: Cambridge University Press, 1985.
Fejzic, H. ``On Thin Sets of Circles.'' Amer. Math. Monthly 103, 582-585, 1996.
Kinney, J. R. ``A Thin Set of Circles.'' Amer. Math. Monthly 75, 1077-1081, 1968.