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 -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), 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