Ramsey–Graham: planar Sidonicity and quasi-independence

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Library card, especially Theorems 3.1, 5.1/5.5, 6.4, and Proposition 7.5.

L. Thomas Ramsey and Colin C. Graham, "Planar Sidonicity and quasi-independence for multiplicative subgroups of the roots of unity," Pacific Journal of Mathematics 225 (2006), no. 2, 325--360.

The paper treats the roots of unity as vectors in the additive plane, decomposes them into prime-coordinate arrays, and develops local criteria for quasi-independence.

Useful machinery

• The square-free theorem reduces (quasi-)independence to intersections with cosets of the square-free part. Theorem 3.1 similarly reduces the Pisier proportional-extraction test to finite subsets inside square-free cosets.
• The spike and shadow lemmas analyze a set one prime-coordinate direction at a time. An empty floor gives an especially clean local-to-global criterion; the later shadow results allow controlled overlaps rather than demanding an empty slice.
• The paper determines or bounds the maximum quasi-independent size $\Psi(n)$ in several families. Its explicit configurations in $T_{105}$, $T_{165}$, and $T_{195}$ are useful finite test cases for any proposed coloring, rank, or circuit argument.

Relation to E0774

The paper supplies a structured finite model in which both extraction density and covering by quasi-independent classes can be studied. It does not itself produce unbounded quasi-independent chromatic number with a uniform extraction constant. Also keep the ambient groups straight: its additive relations are in $\mathbb C$, not in the cyclic exponent group.