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In [3] I showed that there are Helson sets on the circle T which are not of synthesis, by constructing a Helson set which was not of uniqueness and so automatically not of synthesis. In [2] Kaufman gave a substantially simpler construction of such a set; his construction is now standard. It is natural to ask whether there exist Helson sets which are of uniqueness but not of synthesis; this has circulated as an open question. The answer is “yes” and was also given in [3, pp. 87–92] but seems to have got lost in the depths of that rather long paper. Furthermore, the proof depends on the methods of [3], which few people would now wish to master. The object of this note is to give a proof using the methods of [2]. We begin by recalling two definitions.

A Helson set of uniqueness but not of synthesis