Random recursive construction of Salem sets
The Fourier dimension of a compact set \(K \subset \mathbb{R}^d\) with Lebesgue measure \(\lambda^d (K) = 0\), defined by \[ \dim_F (K) : = \sup \biggl\{ \alpha \geq 0\mid\exists\mu\in M_1^+(K):\widehat\mu(x)=O\bigl(|x|^{-\alpha/2}\bigr)\bigl(|x |\to + \infty \bigr) \biggr\}, \] is always majorized by the Hausdorff dimension \(\dim_H (K)\). Here \(M_1^+ (K)\) denotes the set of all probability measures with support in \(K\). \(K\) is called a Salem set of dimension \(\alpha\) if \(\dim_H (K) = \dim_F (K) = \alpha\). In 1950, \textit{R. Salem} [Ark. Mat. 1, 353-365 (1950; Zbl 0054.03001)] proved the existence of linear Salem sets of prescribed dimension in \(]0,1[\) by a rather delicate random dissection method. The random mechanism used by Salem seems only to work in the linear case. In this paper the author introduces a random recursive method for constructing random Salem sets of prescribed dimension in \(\mathbb{R}^d\). Moreover, it is possible to push the topological dimension of the resulting sets down to zero by making them totally disconnected. As an application the author uses the existence of totally disconnected Salem sets of prescribed dimension to prove an existence theorem for sets in \(\mathbb{R}^d\) \((d \geq 2)\) with given topological, Fourier, and Hausdorff dimension.
scientific article; zbMATH DE number 896431