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You have made a few copying errors ($T$ for $T^+$, for instance). For the tiling, my guess is that the claim is not that $\{ T(x) \, : \, x \in \mathbb{Z}^d \}$ is a tiling (that would be false), bu …

Not every distance set $l_k$ will give you a genuine $k$-simplex. But the theorem gives you a realization in $E$ of every set of distances $l_k$, including the ones coming from genuine $k$-simplices. …

Standard apology in case this is something simple, as I'm not a number theorist. Let $E_1, \dots, E_n$ be disjoint finite sets of natural numbers, such that for any $a_1 \in E_1, \dots, a_n \in E_n$, …

Old question and OP probably knows this stuff already, but I thought I'd address it in detail in case someone else comes across it. What you describe is closely related to various notions of irreducib …

Let $A$ be an irreducible nonnegative $N\times N$ integer matrix with constant row sum $D$. Let $A_1, \dots, A_D$ be nonnegative integer matrices, each with constant row sum $1$, such that $\sum_k A_k …

Let $S$ be a finite set and $f: S \to S$ be a function. Let $k = |f(S)|$ and let $\alpha$ be the partition of $S$ into $f$-fibers, i.e. $\alpha = \{ \alpha_t \}_{t \in f(S)}$ where $\alpha_t = f^{-1}( …

This is not really an answer, but it's a suggestion of where to look for a counterexample. I played around for a few hours and couldn't quite get the parameters working, and it's possible they're unwo …