Digital sundials, paradoxical sets, and Vitushkin’s conjecture. (English) Zbl 0609.28005

Falconer, K. J.

Math. Intell. 9, No. 1, 24-27 (1987).

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

Keywords:

orthogonal projection; Hausdorff measure and dimension; Vitushkin’s conjecture; digital sundial; sets with prescribed projections

Citations:

Zbl 0602.28005

References:

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