Massive Helson sets (Q6576766)

A compact subset \(E\) of the torus \(\mathbb T^d\) is called a Helson set if every continuous function on \(E\) can be extended to \(\mathbb T^d\) as a function from the Wiener algebra. Generally, at first glance Helson sets seem to be ``thin''. However, according to the Wik theorem, there exist massive Helson sets on the circle. In particular, they can be of Hausdorff dimension one. In this paper, the Wik theorem is extended to the multidimensional case. Since a Helson set cannot contain a Cartesian product of infinite sets, one cannot obtain a massive Helson set in \(\mathbb T^d\) by just taking the Cartesian product of massive one-dimensional Helson sets. The Helson sets constructed in this paper are Cantor-type sets, i.e., totally disconnected perfect sets.