Random recursive construction of Salem sets
type: Item
instance of: scholarly article
title: Random recursive construction of Salem sets
zbMATH Open document ID: 0851.28005
DOI: 10.1007/BF02559506
author: Christian Bluhm
published in: Arkiv för Matematik
publication date: 1996-11-18T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 28A80
Mathematics Subject Classification ID: 60D05
zbMATH DE Number: 896431
zbMATH Keywords: random fractal measure
zbMATH Keywords: Fourier dimension
zbMATH Keywords: Hausdorff dimension
zbMATH Keywords: Salem set
MaRDI profile type: Publication
cites work: Statistically self-similar fractals
cites work: Brownian Motion and Classical Analysis
cites work: Q3687398
cites work: Descriptive set theory and harmonic analysis
cites work: Q4084463
cites work: Random Recursive Constructions: Asymptotic Geometric and Topological Properties
cites work: Sets of Uniqueness and Sets of Multiplicity. II
cites work: On singular monotonic functions whose spectrum has a given Hausdorff dimension
description: scientific article; zbMATH DE number 896431