The ubiquity of Sidon sets that are not I0

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...

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...

On the ubiquity of Sidon sets

A Sidon set is a set A of integers such that no integer has two essentially distinct representations as the sum of two elements of A. More generally, for every positive integer g, a B2[g]-set is a set A of integers such that no integer has more than g essentially distinct representations as the sum of two elements of A. It is proved that almost all small sumsets of {1, 2, . . . , n} are B2[g]-s...

Sumsets of 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