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Proceedings of the American Mathematical Society

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Random compact sets related to the Kakeya problem


Author: Ralph Alexander
Journal: Proc. Amer. Math. Soc. 53 (1975), 415-419
MSC: Primary 28A75
DOI: https://doi.org/10.1090/S0002-9939-1975-0393427-3
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.


References [Enhancements On Off] (What's this?)

  • [1] A. S. Besicovitch, On Kakeya's problem and a similar one, Math. Z. 27 (1928), 312-320. MR 1544912
  • [2] -, On the fundamental geometrical properties of linearly measurable plane sets of points. III, Math. Ann. 116 (1939), 349-357. MR 1513231
  • [3] -, The Kakeya problem, Amer. Math. Monthly 70 (1973), 697-706. MR 0157266 (28:502)
  • [4] William Feller, An introduction to probability theory and its applications. Vol. I, 2nd ed., Wiley, New York, 1957. MR 19, 466. MR 0088081 (19:466a)

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DOI: https://doi.org/10.1090/S0002-9939-1975-0393427-3
Article copyright: © Copyright 1975 American Mathematical Society

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