The Bose-Chowla argument for 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

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...

Arithmetic Characterizations of Sidon Sets

Let G be any discrete Abelian group. We give several arithmetic characterizations of Sidon sets in G. In particular, we show that a set A is a Sidon set iff there is a number 6 > 0 such that any finite subset A of A contains a subset B Q A with |B| > 6\A\ which is quasiindependent, i.e. such that the only relation of the form ]C\eB e x ^ = '̂ with e\ equal to + 1 or 0, is the trivial one. Let G ...