Sidon Set

A Sidon set is a set of integers for which all sums , with and , are distinct. For a finite Sidon set, this condition is equivalent to requiring that all positive differences between elements be distinct. Subtracting the minimum element from every element therefore turns each finite Sidon set into a Golomb ruler, while every Golomb ruler is a finite Sidon set.

For example, the marks 0, 1, 4, 9, and 11 form a Sidon set and an optimal five-mark Golomb ruler of length 11.

Let denote the largest possible number of elements in a Sidon set contained in the interval , and let denote the minimum length of a -mark Golomb ruler. Then

(1)

Since is known exactly through , this relation determines exactly for (distributed.net 2022).

Singer's (1938) construction using a finite projective plane gives the asymptotic lower bound

(2)

Carter et al. (2025) proved the upper bound

(3)

Hou and Zhao (2026) subsequently improved this result in a preprint to

(4)