Open Sets Avoiding Integral Distances

Abstract

We study open point sets in Euclidean spaces 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 zero. 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. This reveals interesting links between discrete geometry, topology, and measure theory.

Large-volume open sets in normed spaces without integral distances

“…For the Euclidean plane those sets need to have upper density 0, see [1]. Theorem 6.2 proves a conjecture stated in [5] and some of the concepts have been transferred to more general spaces. Nevertheless, many problems remain unsolved and provoke further research.…”

“…We call the just described construction the p-gon construction. These ingredients we provided in more detail in [5] enable us to establish the conjectured exact values of the function…”

“…In [5] the authors provided an example of a connected open set U ⊆ R d such that the intersection of U with each line has a total length of at most 1 but the volume of U is unbounded.…”

“…Here we study a kind of opposite problem, recently considered by the authors for Euclidean spaces, see [5]: Given a normed space X, what is the maximum volume f (X, n) of an open set P ⊆ X with n connected components without a pair of points at integral distance. We drop some technical assumptions for the normed spaces X and mostly consider the Euclidean spaces E d or R d equipped with a p-norm.…”