The distribution of dense Sidon subsets of \({\mathbb Z}_m\) (Q1849623)

Let \(S\) be a Sidon set of residues modulo \(m\). It is shown that the number of elements of \(S\) in any interval of length \(l \) is \( |S|l/m + O(\omega ^{1/2} m^{1/4} \log m)\), where \(\omega = m^{1/2} + 1 - |S|\). This means that Sidon sets near to the maximal possible size are extremely uniformly distributed. To complement this, it is shown that when \(S\) is a perfect difference set, the above discrepancy must be \(\gg m^{1/4}\).