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

DOI: 10.2140/pjm.2006.225.325

Abstract

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

For any set of positive primes, $P$, let $W$ be the be multiplicative subset of $\mathcal{Z}$ generated by $P$. Then $E=\left\{e^{i2\pi a∕m}:a\in \mathcal{Z}\text{ and}m\in W\right\}$ 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, $S\subset e^{2\pi i\mathbb{Q}}$ 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

Milestones

Received: 4 December 2004

Accepted: 12 December 2005

Published: 1 June 2006