We study Sidon and quasi-independence properties (in the discrete complex plane
) 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
For any set of positive primes,
be the be
multiplicative subset of
is a finite union of independent sets (and therefore a Sidon
subset) of the additive group of complex numbers if and only if
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
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.