Bohr Cluster Points of Sidon Sets
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.
- [K] Y. Katznelson, Sequences of integers dense in the Bohr group, in: Proc. Roy. Inst. Techn., June 1973, 73-86.
- [LR] J. M. López and K. A. Ross, Sidon Sets, Marcel Dekker, New York, 1975, pp. 19-52.
- [P] G. Pisier, Arithmetic characterization of Sidon sets, Bull. Amer. Math. Soc. 8 (1983), 87-89.