Eleven Euclidean distances are enough

Abstract

The well-known three-distance theorem states that there are at most three distinct gaps between consecutive elements in the set of fractional parts of the first n multiples of any real number. We generalise this theorem to higher dimensions under a suitable formulation. The three-distance theorem can be thought of as a statement about champions in a tournament. The players in the tournament are geodesics between pairs of multiples of the given real number (modulo 1), two edges play each other if and only if they overlap, and an edge loses only against edges of shorter length that it plays against. According to the three-distance theorem, there are at most three distinct values for the lengths of undefeated edges. In the plane and in higher dimensions, we consider fractional parts of multiples of a vector of real numbers, and two edges play if their projections along any axis overlap. In the plane, there are at most 11 values for the lengths of undefeated edges.