Hostname: page-component-8448b6f56d-t5pn6 Total loading time: 0 Render date: 2024-04-20T06:07:19.600Z Has data issue: false hasContentIssue false
  • English
  • Français

On the size of products of distances from prescribed points

Published online by Cambridge University Press:  24 October 2008

Paul Erdős  and
Vilmos Totik
Paul Erdős
Affiliation:
Mathematical Institute of the Hungarian Academy of Sciences, Budapest, Reáltanoda u. 13–15, 1053, Hungary
Vilmos Totik
Affiliation:
Bolyai Institute, Szeged, Aradi v. tere 1, 6720, Hungary and Department of Mathematics, University of South Florida, Tampa, FL 33620, USA e-mail address: totik@inf.u-szeged.hu
Get access Rights & Permissions [Opens in a new window]

Extract

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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 120 , Issue 3 , October 1996 , pp. 403 - 409
Copyright
Copyright © Cambridge Philosophical Society 1996

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Devore, R. A. and Lorentz, G. G.. Constructive approximation. Grundlehren der Mathematischen Wissenschaften 303 (Springer-Verlag, 1993).CrossRefGoogle Scholar
[2]Fekete, M.. U¨ber den transfiniten Durchmesser ebener Punktmengen. Math. Z. 32 (1930), 108–114.CrossRefGoogle Scholar
[3]Landkof, N. S.. Foundations of modern potential theory. Grundlehren der Mathematischen Wissenschaften 190 (Springer-Verlag, 1972).CrossRefGoogle Scholar
[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.Google Scholar
[5]Tsuji, M.. Potential theory in modern function theory (Maruzen, 1959).Google Scholar
[6]Wagner, G.. On a problem of Erd˝s in diophantine approximation. Bull. London Math. Soc. 12 (1980), 81–88.CrossRefGoogle Scholar