Sidon set

Summary

A Sidon set is a set of integers, or of elements of an abelian group, in which all sums of two elements are different. Here sums that differ only in the order of the summands count as the same. For sets of integers this is the same as saying that all differences of two distinct elements are different. Sidon sets are the extreme opposite of sets with additive structure: their additive energy is as small as possible. The largest Sidon subset of $\set{1, \dots, N}$ has size asymptotic to $\sqrt{N}$.