Abstract

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 = {e^{i2πa ∕ 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 P}1 ∕ p < ∞.

More generally, S ⊂ e^2πiQ 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.

Keywords

independent sets in discrete groups, Sidon sets, quasi-independent sets

Mathematical Subject Classification

Primary: 42A16, 43A46

Secondary: 11A25, 11B99, 11Lxx

Authors