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].
@article{bwmeta1.element.bwnjournal-article-cmv62i1p67bwm,
author = {T. K\"orner},
title = {A Helson set of uniqueness but not of synthesis},
journal = {Colloquium Mathematicae},
volume = {62},
year = {1991},
pages = {67-71},
zbl = {0741.43006},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-cmv62i1p67bwm}
}
Körner, T. A Helson set of uniqueness but not of synthesis. Colloquium Mathematicae, Tome 62 (1991) pp. 67-71. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-cmv62i1p67bwm/
[000] [1] J.-P. Kahane, Séries de Fourier Absolument Convergentes, Springer, 1970. | Zbl 0195.07602
[001] [2] R. Kaufman, M-sets and distributions, Astérisque 5 (1973), 225-230. | Zbl 0281.43006
[002] [3] T. W. Körner, A pseudofunction on a Helson set. I, ibid., 3-224. | Zbl 0281.43004