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On the size of products of distances from prescribed points

  • Paul Erdős (a1) and Vilmos Totik (a2)

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The following problem was raised in the 1993 Miklós Schweitzer Mathematical Contest (see [4]). Let E be any connected set in the plane of diameter greater than 4, and let Z1, Z2, …, be any sequence of points on the plane. Then there is a point X ε E for which infinitely many of the products X¯Z¯1 · X¯Z¯n are greater than 1. Furthermore, the same is not necessarily true if the diameter of E is 4.

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[1]Devore, R. A. and Lorentz, G. G.. Constructive approximation. Grundlehren der Mathematischen Wissenschaften 303 (Springer-Verlag, 1993).
[2]Fekete, M.. U¨ber den transfiniten Durchmesser ebener Punktmengen. Math. Z. 32 (1930), 108114.
[3]Landkof, N. S.. Foundations of modern potential theory. Grundlehren der Mathematischen Wissenschaften 190 (Springer-Verlag, 1972).
[4] Report on the 1993 Miklos Schweitzer Mathematical Contest, Matematikai Lapok (to appear) (in Hungarian with English summary); and Problems in Higher Mathematics II; to be published in the Springer Problem Book Series.
[5]Tsuji, M.. Potential theory in modern function theory (Maruzen, 1959).
[6]Wagner, G.. On a problem of Erd˝s in diophantine approximation. Bull. London Math. Soc. 12 (1980), 8188.

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On the size of products of distances from prescribed points

  • Paul Erdős (a1) and Vilmos Totik (a2)

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