Sidon sequences and nonpositive curvarture

TL;DR

This paper establishes a connection between Sidon sequences and nonpositive curvature by constructing CAT(0) groups and spaces from Sidon sequences, revealing how their arithmetic properties influence geometric structures.

Contribution

It introduces a novel method to construct CAT(0) spaces from Sidon sequences and links their arithmetic properties to geometric curvature and flat plane structures.

Findings

Sidon sequences are equivalent to nonpositive curvature conditions.

The number of representations as alternating sums determines flat plane structures.

Constructed CAT(0) spaces exhibit properties derived from Sidon sequence arithmetic.

Abstract

A sequence $a_0<a_1<\ldots<a_n$ 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.