Ramsey–Graham: permutation and extension

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Library card, especially Theorems 1.2.1 and 1.2.2, Corollary 2.1.3, and the extension theorem in Section 4.

L. Thomas Ramsey and Colin C. Graham, "Permutation and extension for planar quasi-independent subsets of the roots of unity," arXiv:math/0606546 (2006).

For odd square-free $n=\prod p_i$, the coordinatewise permutations of $\prod \mathbb Z_{p_i}$ are exactly the permutations preserving the class of quasi-independent sets (and also the class of independent sets). This gives a large symmetry group for normalizing finite configurations. The empty-floor criterion and extension results show how a quasi-independent configuration survives when a prime coordinate is enlarged or a new prime factor is added.