A Proof That the Union of Two Helson Sets Is a Helson Set

Abstract

The question, whether the union of two Helson sets is a Helson set, resisted answering for some time. S. W. Drury and N. Th. Varopoulos solved the problem in 1970, and we now know that if H = H₁ ∪ H₂ where H₁ and H₂ are Helson subsets of G, then

$$\alpha \left( H \right) \leqslant \frac{{{3^{{3/2}}}}}{2}(\alpha {({H_{1}})^{3}} + \alpha {({H_{2}})^{3}}). $$

One may still hope for simpler proofs and better inequalities.