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Some remarks on the Kakeya problem

  • Roy O. Davies (a1)
Abstract

Besicovitch's construction(1) of a set of measure zerot containing an infinite straight line in every direction was subsequently adapted (2, 3, 4) to provide the following answer to Kakeya's problem (5): a unit segment can be continuously turned round, so as to return to its original position with the ends reversed, inside an arbitrarily small area. The last word on Kakeya's problem itself seems to be F. Cunningham Jr.'s remarkable result(6)‡ that this can be done inside a simply connected subset of arbitrarily small measure of a unit circle.

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(1)Besicovitch A. S.Sur deux questions de l'intégrabiité des fonctions. J. Soc. Phys. Math. (Perm) 2 (1919), 105123.
(2)Besicovitch A. S.On Kakeya's problem and a similar one. Math. Z. 27 (1928), 312320.
(3)Perron O.Über emen Satz von Besicovitch. Math. Z. 28 (1928), 383386.
(4)Besicovitch A. S.The Kakeya problem. Amer. Math. Monthly 70 (1963), 697706.
(5)Kakeya S.Some problems on maxima and minima regarding ovals. Tôhoku Sci. Reports 6 (1917), 7188.
(6)Cunningham F. Jr. The Kakeya problem for simply-connected and for star-shaped sets, to be published.
(7)Besicovitch A. S. and Rado R.A piano set of measure zero containing circumferences of every radius. J. London Math. Soc. 43 (1968), 717719.
(8)Kinney J. R.A thin set of circles. Amer. Math. Monthly 75 (1968), 10771081.
(9)Davies Roy O., Marstrand J. M. and Taylor S. J.On the intersections of transforms of linear sets. Colloq. Math. 7 (1960), 237243.
(10)Ward J. A set of plane measure zero containing all finite polygonal arcs. Caned. J. Math., forthcoming.
(11)Ward D. J. Some dimensional properties of generalised difference sets. Mathematika, forthcoming.
(12)Marstrand J. M.Some fundamental geometrical properties of plane sets of fractional dimensions. Proc. London Math. Soc. (3) 4 (1954), 257302.
(13)Marstrand J. M.The dimension of Cartesian product sets. Proc. Cambridge Philos. Soc. 50 (1954), 198202.
(14)Besicoviron A. S.On fundamental geometric properties of plane line-sets. J. London Math. Soc. 39 (1964), 441448.
(15)Croft H. T.Review of (14). Math. Rev. 30 (1965), no. 2122.
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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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