## Spectra of Bernoulli convolutions as multipliers in $$L^p$$ on the circle.(English)Zbl 1059.47004

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 $${\mathcal F}^{(j)}_{\mu_{\theta^r}}$$.

### MSC:

 47A10 Spectrum, resolvent 42A16 Fourier coefficients, Fourier series of functions with special properties, special Fourier series 11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure
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