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The dimension of Cartesian product sets

  • J. M. Marstrand (a1)
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Given a plane set E, we denote by Ex the set of its points whose abscissae are equal to x.

Throughout this paper we use the letters A and B to denote subsets of the x-axis and y-axis respectively, and we denote by A × B their Cartesian product set. We use the letters s and t to denote positive numbers; we denote by ΛsE the outer Hausdorff s-dimensional measure of the set E.

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(1)Besicovitch, A. S.On existence of subsets of finite measure of sets of infinite measure. Indag. math. 14 (1952), 339–44.
(2)Besicovitch, A. S. and Moran, P. A. P.The measure of product and cylinder sets. J. Lond. math. Soc. 20 (1945), 110–20.
(3)Davies, R. O.Subsets of finite measure in analytic sets. Indag. math. 14 (1952), 488–9.
(4)Eggleston, H. G.The Besicovitch dimension of cartesian product sets. Proc. Camb. phil. Soc. 46 (1950), 383–6.
(5)Eggleston, H. G.A correction to a paper on the dimension of cartesian product sets. Proc. Camb. phil. Soc. 49 (1953), 437–40.
(6)Moran, P. A. P.On plane sets of fractional dimensions. Proc. Lond. math. Soc. 51 (1949), 415–23.
(7)Saks, S.Theory of the Integral (Warsaw, 1937).
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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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