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

摘要

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) $ .