Let μθ,D\documentclass[12pt]{minimal}
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\begin{document}$$\mu _{\theta ,D}$$\end{document} be the self-similar measure on R\documentclass[12pt]{minimal}
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\begin{document}$${{\mathbb {R}}}$$\end{document} satisfying that μθ,D:=1#D∑j∈Dμθ,D∘φj-1\documentclass[12pt]{minimal}
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\begin{document}$$\mu _{\theta ,D}:=\frac{1}{\#D} \sum _{j \in D} \mu _{\theta ,D}\circ \varphi _j^{-1}$$\end{document}, where φj(x)=θ-1x+j,θ>1,j∈D⊆Z,D=-D,\documentclass[12pt]{minimal}
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\begin{document}$$\varphi _j(x)=\theta ^{-1}x+j, \ \theta >1, j\in D\subseteq {{\mathbb {Z}}}, D=-D,$$\end{document} and #D\documentclass[12pt]{minimal}
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\begin{document}$$\#D$$\end{document} denotes the cardinality of the set D. In this work, we will show that, under a mild condition, the closure {μθ,D^(rn):n∈Z}¯\documentclass[12pt]{minimal}
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\begin{document}$$\overline{\{\widehat{\mu _{\theta ,D}}(rn):n \in {{\mathbb {Z}}}\}}$$\end{document} of the set of Fourier transforms {μθ,D^(rn):n∈Z}\documentclass[12pt]{minimal}
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\begin{document}$$\{\widehat{\mu _{\theta ,D}}(rn):n \in {{\mathbb {Z}}}\}$$\end{document} of the self-similar measure μθ,D\documentclass[12pt]{minimal}
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\begin{document}$$\mu _{\theta ,D}$$\end{document} parameterized by a Pisot number θ\documentclass[12pt]{minimal}
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\begin{document}$$\theta $$\end{document} is countable for all positive r∈Q(θ)\documentclass[12pt]{minimal}
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\begin{document}$$r\in {{\mathbb {Q}}}(\theta )$$\end{document} but uncountable for Lebesgue-a.e. r>0\documentclass[12pt]{minimal}
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\begin{document}$$r>0$$\end{document}. As an application, this, together with results of Sarnak [19] and Hu [8], proves that, for every fixed θ>1\documentclass[12pt]{minimal}
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\begin{document}$$\theta >1$$\end{document} and the digit set D which is either D=±{0,1,…,q}\documentclass[12pt]{minimal}
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\begin{document}$$D=\pm \{0,1,\ldots ,q\}$$\end{document} or D=±{1,3,…,2q-1}\documentclass[12pt]{minimal}
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\begin{document}$$D=\pm \{1,3,\ldots ,2q-1\}$$\end{document} where q∈N\documentclass[12pt]{minimal}
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\begin{document}$$q\in {{\mathbb {N}}}$$\end{document}, the spectrum of the convolution operator f↦μθ,D∗f\documentclass[12pt]{minimal}
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\begin{document}$$f\mapsto \mu _{\theta ,D}*f$$\end{document} in Lp(T)\documentclass[12pt]{minimal}
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\begin{document}$$L^p(\mathbb {T})$$\end{document} (where T\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {T}$$\end{document} is the circle group) is countable and is the same for all p∈(1,∞)\documentclass[12pt]{minimal}
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\begin{document}$$p\in (1,\infty )$$\end{document}, that is, {μθ,D^(n):n∈Z}¯\documentclass[12pt]{minimal}
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\begin{document}$$\overline{\{\widehat{\mu _{\theta ,D}}(n):n \in {{\mathbb {Z}}}\}}$$\end{document}. This extends the corresponding results of Erdös [6], Salem [17], and Sidorov and Solomyak [21] for Bernoulli convolutions.
