×

An extremal set of uniqueness? (English) Zbl 0838.43006

Let \(S\) be the set of all dyadic rational numbers \(x= \sum^\infty_{k= 1} {x^k\over 2^k}\), \(x_k\in \{0, 1\}\) with \(x_k= 0\), \(k< \sum_k x_k\) (including \(x= 0\)). As a countable closed set (in \(\mathbb{R}/\mathbb{Z})\), \(S\) is a set of uniqueness. The authors show that \(S\) cannot be expressed as the union of two \(H\)-sets. \(E\) is an \(H\)-set if there exists a nonempty open interval \(I\subseteq \mathbb{R}/\mathbb{Z}\) such that \(nE\cap I\) is void for infinitely many \(n\in \mathbb{Z}\). They conjecture that \(S\) is an extremal set of uniqueness, i.e. \(S\) is not the union of a finite number of \(H\)-sets, and prove a weaker result: \(S\) has infinite (Cantor-Bendixson) rank.
Reviewer: H.Rindler (Wien)

MSC:

43A46 Special sets (thin sets, Kronecker sets, Helson sets, Ditkin sets, Sidon sets, etc.)
PDFBibTeX XMLCite
Full Text: DOI EuDML