Random compact sets related to the Kakeya problem
Proc. Amer. Math. Soc. 53 (1975), 415-419
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Abstract: A -set is defined to be a compact planar set of zero measure which contains a translate of any line segment lying in a disk of diameter one. A construction is given which associates a unique compact planar set with each sequence in a closed interval, and it is shown that for almost all such sequences a -set is obtained. The construction depends on the measure properties of certain perfect linear sets. Several related problems of a subtler nature are also considered.
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