Spectra of Bernoulli convolutions as multipliers in \(L^p\) on the circle (Q1423877)

If \(\mathbb{Z}\), \(\mathbb{Q}\), \(\mathbb{R}\), denote the set of integers, rational numbers and real numbers, respectively, let \(S^1\) denote \(\mathbb{R}\setminus \mathbb{Z}\), and if \(v\) is a Borel probability measure on \(S^1\), let \(T_v: L^p(S^1)\to L^p(S^1)\) denote the convolution operator defined by \(T_v(f)= v*f\). The set of Fourier coefficients \(\{v^\wedge(n),\,n\in \mathbb{Z}\}\) is denoted by \({\mathcal F}_v\) and \(\{v^\wedge(rn),\,n\in \mathbb{Z}\}\) is denoted by \({\mathcal F}_{v,r}\). If \(\theta\) is a Pisot number, so that \(\theta> 1\) and has conjugates with modulus less than 1, then the Bernoulli convolution \(\mu_\theta\) is defined by \[ \begin{gathered}\mu_\theta= \Pi^*\{1/2 \delta_{-\theta^{-k}}+ 1/2\delta_{\theta^{-k}},\, k= 0,1,2,\dots\},\\ \widehat\mu_\theta(t)= \Pi\{\cos(2\pi\theta^{- k}t),\,k= 0,1,2,\dots\}.\end{gathered} \] In one of the main results of this paper, the authors show that if \(\theta\neq 2\) is a Pisot number and if \(r\in Q(\theta)\), then the set \({\mathcal F}_{\mu_{\theta^r}}'\) of limit points of \({\mathcal F}_{\mu_{\theta^r}}\) is infinite and countable. In addition, for Lebesgue almost all \(r> 0\), the set \({\mathcal F}_{\mu_{\theta^r}}'\) is a non-empty interval. The main conclusions are still applicable if \({\mathcal F}_{\mu_{\theta^r}}'\) is replaced by \({\mathcal F}^{(n)'}_{\mu_{\theta^r}}\), where \({\mathcal F}^{(j+1)'}_{\mu_{\theta^r}}\) is the set of limit points of \({\mathcal F}^{(j)}_{\mu_{\theta^r}}\).