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Digital sundials, paradoxical sets, and Vitushkin’s conjecture. (English) Zbl 0609.28005
A digital sundial is an ”object” whose shadow at any given time shows the digits of that time. The existence of such a magic clock is a consequence of a more general result of the author on finding sets with prescribed projections [Proc. Lond. Math. Soc., III. Ser. 53, 48-64 (1986; Zbl 0602.28005)]. In this article the author discusses the underlying construction, earlier related results of Besicovitch, Davies and Marstrand, and a recent use of such constructions by the reviewer to disprove a conjecture in complex function theory.
Reviewer: P.Mattila

MSC:
28A75 Length, area, volume, other geometric measure theory
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[1] A. S. Besicovitch, ”On the fundamental geometrical properties of linearly measurable sets of points”,Math Ann, 98 (1928), 422–464. · JFM 53.0175.04 · doi:10.1007/BF01451603
[2] R. O. Davies, ”On accessibility of plane sets and differentiation of functions of two real variables”,Proc Cambridge Philos Soc, 48 (1952), 215–232. · Zbl 0046.05801 · doi:10.1017/S0305004100027584
[3] K. J. Falconer, The Geometry of Fractal Sets, Cambridge University Press, 1985. · Zbl 0587.28004
[4] K. J. Falconer, ”Sets with prescribed projections and Ni-kodym sets”,Proc London Math Soc (3),53 (1986), 48–64. · Zbl 0602.28005 · doi:10.1112/plms/s3-53.1.48
[5] J. M. Marstrand, ”Some fundamental geometric properties of plane sets of fractional dimensions”,Proc London Math Soc (3),4 (1954), 257–302. · Zbl 0056.05504 · doi:10.1112/plms/s3-4.1.257
[6] P. Mattila, ”Smooth maps, null-sets for integralgeometric measure and analytic capacity”,Ann Math, 123 (1986), 303–309. · Zbl 0589.28006 · doi:10.2307/1971273
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