Generalized Sidon sets

Generalized Sidon sets

We give asymptotic sharp estimates for the cardinality of a set of residue classes with the property that the representation function is bounded by a prescribed number. We then use this to obtain an analogous result for sets of integers, answering an old question of Simon Sidon. © 2010 Elsevier Inc. All rights reserved. MSC: 11B

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 = s1 + s2, si ∈ S; such sets are called Sidon sets if g = 2 and generalized Sidon sets (or B2[ ⌈ g/2 ⌉ ] sets) if g ≥ 3. We extend to generalized Sidon sets the Sidon-set constructions of Singer, Bose, and Ruzsa. We also further optimize Koulantzakis’ idea of interleaving sever...

Bounds for generalized Sidon sets

Let Γ be an abelian group and g ≥ h ≥ 2 be integers. A set A ⊂ Γ is a Ch[g]-set if given any set X ⊂ Γ with |X | = h, and any set {k1, . . . , kg } ⊂ Γ , at least one of the translates X + ki is not contained in A. For any g ≥ h ≥ 2, we prove that if A ⊂ {1, 2, . . . , n} is a Ch[g]-set in Z, then |A| ≤ (g − 1)1/hn1−1/h + O(n1/2−1/2h). We show that for any integer n ≥ 1, there is a C3[3]-set A ...

Sumsets of Sidon sets

Perfect difference sets constructed from Sidon sets

A set A of positive integers is a perfect difference set if every nonzero integer has an unique representation as the difference of two elements of A. We construct dense perfect difference sets from dense Sidon sets. As a consequence of this new approach we prove that there exists a perfect difference set A such that A(x) ≫ x √ . Also we prove that there exists a perfect difference set A such t...