Beurling dimension and self-similar measures
The authors prove that the following conjecture is true for Bessel sets of a self-similar measure \(\mu\) on \(\mathbb{R}^d\) which satisfies the open set condition and has different contraction ratios and different probability weights: Conjecture. Let \(\mu\) be a Borel probability measure with compact support \(T\subset\mathbb{R}^d\). If \(\mu\) is a frame spectral measure with frame spectrum \(\Lambda\) then \[ \dim_B \Lambda \leq \dim_H T, \] where \(\dim_B\) denotes the Beurling and \(\dim_H\) the Hausdorff dimension. Several examples are given to illustrate the result.
scientific article; zbMATH DE number 6845288
scientific article