$k$-Fold Sidon Sets

k-Fold Sidon Sets

Let k > 1 be an integer. A set A ⊂ Z is a k-fold Sidon set if A has only trivial solutions to each equation of the form c1x1 + c2x2 + c3x3 + c4x4 = 0 where 0 6 |ci| 6 k, and c1 + c2 + c3 + c4 = 0. We prove that for any integer k > 1, a k-fold Sidon set A ⊂ [N ] has at most (N/k)1/2 + O((Nk)1/4) elements. Indeed we prove that given any k positive integers c1 < · · · < ck, any set A ⊂ [N ] that c...

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

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 ...