  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 zero. We are interested in
how large such sets can be in $d$-dimensional volume. We determine the lower
and upper bounds for the volumes of the sets in terms of the number of their
connected components and dimension, and also give some exact values. Our
problem can be viewed as a kind of inverse to known problems on sets with
pairwise rational or integral distances.
