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Some remarks on the Kakeya problem

Published online by Cambridge University Press:  24 October 2008

Roy O. Davies
Affiliation:
The University, Leicester

Extract

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.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1971

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References

REFERENCES

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