The distribution of dense Sidon subsets of $ \mathbb{Z}_m $

Abstract.

Let $S \subseteqq \mathbb{Z}_m$ be a Sidon set of cardinality $\mid S \mid = m^{1 \over 2} + O(1)$. It is proved, in particular, that for any interval ${\cal I} = \{a, a + 1, \ldots, a + \ell - 1\}$ in $\mathbb{Z}_m$, $0 \leqq \ell$ < m, we have $\big| {\mid S \cap {\cal I} \mid - \mid S \mid \ell/m} \big| = O(\mid S \mid^{1 \over 2}\textrm{ln}\, m)$.