Discrete & Computational Geometry

, Volume 50, Issue 1, pp 99–123 | Cite as

Open Sets Avoiding Integral Distances

Article

Abstract

We study open point sets in Euclidean spaces \(\mathbb{R }^d\) without a pair of points an integral distance apart. By a result of Furstenberg, Katznelson, and Weiss such sets must be of Lebesgue upper density \(0\). We are interested in how large such sets can be in \(d\)-dimensional volume. We determine the exact values for the maximum volumes of the sets in terms of the number of their connected components and dimension. Here techniques from diophantine approximation, algebra and the theory of convex bodies come into play. Our problem can be viewed as a counterpart to known problems on sets with pairwise rational or integral distances. It possibly opens a new research direction with strong links to topology and measure theory.

Keywords

Excluded distances Euclidean Ramsey theory Integral distances Erdős-type problems 

Mathematics Subject Classification

52C10 52A40 51K99 

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of BayreuthBayreuthGermany
  2. 2.Department of Mathematics and StatisticsYork UniversityTorontoCanada

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