Open Sets Avoiding Integral Distances
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Open Sets Avoiding Integral Distances Sascha Kurz · Valery Mishkin
Received: 2 April 2012 / Revised: 13 March 2013 / Accepted: 22 April 2013 / Published online: 23 May 2013 © Springer Science+Business Media New York 2013
Abstract We study open point sets in Euclidean spaces Rd 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 · Erd˝os-type problems Mathematics Subject Classification
52C10 · 52A40 · 51K99
1 Introduction Is there a dense set S in the plane so that all pairwise Euclidean distances between the points are rational? This famous open problem was posed by Ulam in 1945, see e.g. [18,39]. Unlike this, a construction of a countable dense set in the plane avoiding
S. Kurz (B) Department of Mathematics, University of Bayreuth, 95440 Bayreuth, Germany e-mail: [email protected] V. Mishkin Department of Mathematics and Statistics, York University, Toronto, ON M3J1P3, Canada e-mail: [email protected]
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rational distances is not hard to find, see e.g. [27, Problems 13.4, 13.9]. If all pairwise distances between the points in S are integral and S is non-collinear, i.e. not all points are located on a line, then S is finite [2,17]. Having heard of this result, Ulam guessed that the answer to his question would be in the negative. Of course the rational numbers form a dense subset of a coordinate line with pairwise rational distances; also, on a circle there are dense sets with pairwise rational distances, see e.g. [1,2]. It was proved by Solymosi and De Zeeuw [37] that the line and the circle are the only two irreducible algebraic curves containing infinite subsets of points with pairwise rational distances. Point sets with rational coordinates on spheres have been considered in [34]. There is interest in a general construction of a planar point set S(n, k) of size n with pairwise integral distances such that S(n, k) = A ∪ B where A is collinear, |A| = n − k, |B| = k, and B has no three collinear points. The current record is k = 4 [11]. And indeed, it is very hard to construct a planar point set, no three points on a line, no four points on a circle, with pairwise integral distances. Kreisel and Kurz [28] found such a set of size 7, but it is unknown if there exists one of size 8. The present paper is concerned with a problem that may be considered as a counterpart t
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