Sidon sets and $I^0$-sets

Comparisons of Sidon and I0 Sets

(1) Bd(E) and B(E) are isometrically isomorphic for finite E ⊂ Γ. Bd(E) = `∞(E) characterizes I0 sets E and B(E) = `∞(E) characterizes Sidon sets E. [In general, Sidon sets are distinct from I0 sets. Within the group of integers Z, the set {2}n ⋃ {2+n}n is helsonian (hence Sidon) but not I0.] (2) Both are Fσ in 2 (as is also the class of finite unions of I0 sets). (3) There is an analogue for I...

Proportions of Sidon Sets Are I0 Subsets

It is proved that proportions of Sidon sets are I0 subsets of controlled degree. That is, a set E is Sidon if and only if, there are r > 0 and positive integer n such that, for every finite subset F ⊂ E, there is H ⊂ F with the cardinality of H at least r times the cardinality of F and N(H) ≤ n (N(H) is a measure of the degree of being I0). This paper leaves open David Grow’s question of whethe...

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...

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