Sidon sequences and nonpositive curvature

Abstract

A sequence $a_0<a_1<\cdots <a_n$ of nonnegative integers is called a Sidon sequence if the sums of pairs $a_i+a_j$ are all different. In this paper we construct $CAT(0)$ groups and spaces from Sidon sequences. The arithmetic condition of Sidon is shown to be equivalent to nonpositive curvature, and the number of ways to represent an integer as an alternating sum of triples $a_i-a_j+a_k$ of integers from the Sidon sequence, is shown to determine the structure of the space of embedded flat planes in the associated $CAT(0)$ complex.