Separated centers under finitely many torus covers

Statement

Let $J$ be a nonsingular integer matrix and $D\ge0$ an integer. Assume $J^j-I$ is nonsingular for $1\le j\le D$ . For every finite number $N$ of labels, there are rational torus points $c_1,\ldots,c_N$ and $r>0$ such that

J^j\overline B(c_\nu,r)\cap\overline B(c_\mu,r)=\varnothing

for all $0\le j\le D$ , except $j=0,\mu=\nu$ . Images and balls are taken on the unit flat torus.

Proof

Choose the centers to avoid $J^jc_\nu=c_\mu$ . For distinct labels this equation is a closed set with empty interior in the product of tori. For the same label and $j>0$ , the forbidden points form the finite kernel of the covering induced by $J^j-I$ . The complement of the finitely many forbidden sets is open and dense; rational tuples are dense, so a rational tuple lies in the complement.

The finitely many nonzero distances $d(J^jc_\nu,c_\mu)$ have positive minimum $\eta$ . Choose $r$ with $(1+\max_{0\le j\le D}\|J^j\|)r<\eta$ . The triangle inequality gives the result. It follows that corresponding preimages at levels whose difference is at most $D$ are disjoint, for every nonexcluded label pair.