Abstract
Let \(\left\{ {{d_k}} \right\}_{k = 1}^\infty \) be an upper-bounded sequence of positive integers and let δE be the uniformly discrete probability measure on the finite set E. For 0 < ρ < 1, the infinite convolution \({\mu _{\rho \left\{ {0,{d_k}} \right\}}}: = {\delta _{\rho \left\{ {0,{d_1}} \right\}}}*{\delta _{{\rho ^2}\left\{ {0,{d_2}} \right\}}}* \cdots \) is called an infinite Bernoulli convolution. The non-spectral problem on \({\mu _{\rho ,\left\{ {0,{d_k}} \right\}}}\) is to investigate the cardinality of orthogonal exponentials in \({L^2}\left( {{\mu _{\rho ,\left\{ {0,{d_k}} \right\}}}} \right)\). In this paper, we give a characterization of this problem by classifying the values of ρ.
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The authors would like to thank the referee for careful reading and the valuable suggestions and express the same thanks to the editor for careful proofreading and editing.
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This work was supported by NSFC grants (11771457, 11971194), Guangdong Province Key Laboratory of Computational Science at the Sun Yat-sen University and by the Foundation of China Postdoctoral Science (2020M672928).
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Li, Q., Wu, ZY. Non-Spectral Problem on Infinite Bernoulli Convolution. Anal Math 47, 343–355 (2021). https://doi.org/10.1007/s10476-021-0069-7
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DOI: https://doi.org/10.1007/s10476-021-0069-7