Vol. 225, No. 2, 2006

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ISSN: 0030-8730
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 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 = {ei2πam : a Z andm W} is a finite union of independent sets (and therefore a Sidon subset) of the additive group of complex numbers if and only if pP1p < .

More generally, S e2πi 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
Authors
L. Thomas Ramsey
Department of Mathematics
University of Hawaii
Keller Hall
2565 The Mall
Honolulu, HA 96822
United States
Colin C. Graham
1115 Lenora Road
Bowen Island BC V0N 1G0
Canada