On self-similar spectral measures (Q2215828)

The authors study the spectral properties of self-similar measures \(\mu_{\rho,\mathcal{D}}\) on the real line associated with iterated function systems of the form \(\{\tau_d = \rho(\cdot + d)\}_{d\in \mathcal{D}}\) where \(0 < \rho < 1\) and \(\mathcal{D} \subset \mathbb{R}\) is a finite digit set. Such measures can be written as \[ \mu_{\rho,\mathcal{D}} = \delta_{\rho\mathcal{D}} * \delta_{\rho^2\mathcal{D}} * \delta_{\rho^3\mathcal{D}} * \cdots \] \[ = \mu_k * \mu_{\rho\mathcal{D}}(\rho^{-k}), \] where \(\delta_x\) denotes the point measure at \(x\) and \(\mu_k\) the convolution product of the first \(k\) discrete measures. It is shown that the conditions (i) \(\rho^-1\in \mathbb{N}\) and (ii) \(\mu_k\) is a spectral measure for all \(k\in \mathbb{N}\) are necessary -- under some natural assumptions -- for \(\mu_{\rho,\mathcal{D}}\) to be a spectral measure. As an application of this result, a self-similar spectral measure associated with an integer tile is characterized.