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 -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.

**[1]**A. S. Besicovitch,*On Kakeya’s problem and a similar one*, Math. Z.**27**(1928), no. 1, 312–320. MR**1544912**, 10.1007/BF01171101**[2]**A. S. Besicovitch,*On the fundamental geometrical properties of linearly measurable plane sets of points (III)*, Math. Ann.**116**(1939), no. 1, 349–357. MR**1513231**, 10.1007/BF01597361**[3]**A. S. Besicovitch,*The Kakeya problem*, Amer. Math. Monthly**70**(1963), 697–706. MR**0157266****[4]**William Feller,*An introduction to probability theory and its applications. Vol. I*, John Wiley and Sons, Inc., New York; Chapman and Hall, Ltd., London, 1957. 2nd ed. MR**0088081**

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DOI:
https://doi.org/10.1090/S0002-9939-1975-0393427-3

Article copyright:
© Copyright 1975
American Mathematical Society