Infinite rational distance sets in affine general position: constructions in every dimension

Abstract

For every integer $d\ge 1$, we construct a countably infinite set $X_d\subset {\mathbb{R}}^d$ in affine general position, with all pairwise distances rational. When $d$ is odd, $X_d$ may also be chosen so that no $d+2$ points lie on a common sphere. The construction is uniform in $d$: positive Chebyshev square decompositions produce harmonic curves on spheres whose points corresponding to rational parameter values have pairwise rational distances. A divided-difference factorization of the affine determinant shows that sufficiently short arcs are locally convex, and stereographic projection produces the odd-dimensional examples. We also construct infinite rational distance sets in ${\mathbb{Q}}^d$ in affine general position for every even $d$, and in general position for every $d\equiv 1(\mathrm{mod}4)$. For every $d\ge 1$ and $n\ge d+1$, taking and rescaling suitable finite subsets gives $n$-point integral point sets in affine general position. A suitable ordered choice yields integral-distance realizations of all cyclic polytopes. In dimension three, we give an explicit rational parametrization and obtain infinitely many pairwise non-similar primitive $n_3$-clusters for every $n\ge 4$.

27 pages; a 7-page computational supplement and exact verification code are included as ancillary files