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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Random compact sets related to the Kakeya problem


Author: Ralph Alexander
Journal: Proc. Amer. Math. Soc. 53 (1975), 415-419
MSC: Primary 28A75
MathSciNet review: 0393427
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Abstract: A $ B$-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 $ B$-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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DOI: http://dx.doi.org/10.1090/S0002-9939-1975-0393427-3
PII: S 0002-9939(1975)0393427-3
Article copyright: © Copyright 1975 American Mathematical Society