Bohr Cluster Points of Sidon Sets

If there is a Sidon subset of the integers Z which has a member of Z as a cluster point in the Bohr compactification of Z, then there is a Sidon subset of Z which is dense in the Bohr compactification. A weaker result holds for quasiindependent and dissociate subsets of Z. It is a long standing open problem whether Sidon subsets of Z can be dense in the Bohr compactification of Z ([LR]). Yitzhak Katznelson came closest to resolving the issue with a random process in which almost all sets were Sidon and and almost all sets failed to be dense in the Bohr compactification [K]. This note, which does not resolve this open problem, supplies additional evidence that the problem is delicate: it is proved here that if one has a Sidon set which clusters at even one member of Z, one can construct from it another Sidon set which is dense in the Bohr compactification of Z. A weaker result holds for quasi-independent and dissociate subsets of Z. Cluster Points. By the definition of the Bohr topology, a subset E ⊂ Z clusters at q if and only if, for all ∈ R, for all n ∈ Z, and for all (t1, . . . , tn) ∈ T, there is some m ∈ E such that (1) sup 1≤i≤n | < m, ti > − < q, ti > | < . Here T is the dual group of Z and < m, t > denotes the result of the character m acting on t. Thus, if T is represented as [−π, π) with addition mod 2π, < m, t >= e. If, for all (t1, . . . , tn) ∈ T, there is at least one m ∈ E such that inequality (1) holds, then E is said to approximate q within on T. 1991 Mathematics Subject Classification. 43A56.