Constructions of generalized Sidon sets

We give explicit constructions of sets S with the property that for each integer k , there are at most g solutions to k = s 1 + s 2 , s i ∈ S ; such sets are called Sidon sets if g = 2 and generalized Sidon sets if g ⩾ 3 . We extend to generalized Sidon sets the Sidon-set constructions of Singer, Bose, and Ruzsa. We also further optimize Kolountzakis’ idea of interleaving several copies of a Sidon set, extending the improvements of Cilleruelo, Ruzsa and Trujillo, Jia, and Habsieger and Plagne. The resulting constructions yield the largest known generalized Sidon sets in virtually all cases.