Vol. 225, No. 2, 2006

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

L. Thomas Ramsey and Colin C. Graham

Vol. 225 (2006), No. 2, 325–360
Abstract

We study Sidon and quasi-independence properties (in the discrete complex plane $\mathsc{C}$) 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 $\mathsc{C}$.

For any set of positive primes, $P$, let $W$ be the be multiplicative subset of $\mathsc{Z}$ generated by $P$. Then is a finite union of independent sets (and therefore a Sidon subset) of the additive group of complex numbers if and only if ${\sum }_{p\in P}1∕p<\infty$.

More generally, $S\subset {e}^{2\pi iℚ}$ 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 2000
Primary: 42A16, 43A46
Secondary: 11A25, 11B99