Sidon set

Remarks

If $G$ has no elements of order $2$, as in $\Z$, the definition is equivalent to this one: the differences $a - b$ with $a, b \in A$ and $a \ne b$ are pairwise different. The set $\set{1, 2, 4, 8}$ is a Sidon set in $\Z$. Its sums $a + b$ with $a \le b$ are $2, 3, 4, 5, 6, 8, 9, 10, 12, 16$, and these ten numbers are distinct. The set $\set{1, 2, 3, 4}$ is not a Sidon set, since $1 + 4 = 2 + 3$ but $\set{1, 4} \ne \set{2, 3}$. An infinite Sidon set of positive integers, listed in increasing order, is called a Sidon sequence. Sidon sets in $\Z$ are also called Golomb rulers when they are finite.