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 I0 sets of the sup-norm partition construction used with Sidon sets. (4) A set E is Sidon if and only if, there is some r ∈ R and positive integer N such that, for all finite F ⊂ E, there is some H ⊂ F with |H| ≥ r|F | and H is an I0 set of degree at most N . [Here |S| denotes the cardinality of S; two different but comparable definitions of degree for I0 sets are made below.] (5) IF all Sidon subsets of Z are finite unions of I0 sets, the number of I0 sets required is bounded by some function of the Sidon constant. This is also true in the category of all discrete abelian groups.