A Helson set of uniqueness but not of synthesis
In  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  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 , which few people would now wish to master. The object of this note is to give a proof using the methods of .
-  J.-P. Kahane, Séries de Fourier Absolument Convergentes, Springer, 1970.
-  R. Kaufman, M-sets and distributions, Astérisque 5 (1973), 225-230.
-  T. W. Körner, A pseudofunction on a Helson set. I, ibid., 3-224.