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We study Sidon and quasi-independence properties (in the discrete complex plane C) for subsets of the roots of unity. We obtain criteria for sets of roots of unity to be quasi-independent and to be Sidon in C.For any set of positive primes, P, let W be the be multiplicative subset of Z generated by P. Then E = {ei2pa / m : a in Z and m in W} is a finite union of independent sets (and therefore a Sidon subset) of the additive group of complex numbers if and only if ? p in P1 / p < 8.More generally, S ? e2piQ is a Sidon set if and only if its intersections with cosets of certain (multiplicative) subgroups, those with square-free order, satisfy a (quasi-independence related) criterion of Pisier.Certain new aspects of the combinatorial geometry of the integer-coordinate points in n-dimensional Euclidean space are shown to be equivalent to quasi-independence for subsets of the roots of unity. These aspects are fully resolved in two-dimensional Euclidean space but lead to combinatorial explosion in three dimensions.

Planar Sidonicity and quasi-independence for multiplicative subgroups of the roots of unity