Let $$\left\{ {{d_k}} \right\}_{k = 1}^\infty $$ be an upper-bounded sequence of positive integers and let δ E 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 .

中文翻译：

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无限伯努利卷积的非谱问题

令 $$\left\{ {{d_k}} \right\}_{k = 1}^\infty $$ 是一个上界的正整数序列，让 δ E 是有限集上的均匀离散概率测度乙。对于 0 < ρ < 1，无限卷积 $${\mu _{\rho \left\{ {0,{d_k}} \right\}}}: = {\delta _{\rho \left\{ { 0,{d_1}} \right\}}}*{\delta _{{\rho ^2}\left\{ {0,{d_2}} \right\}}}* \cdots $$ 称为无穷大伯努利卷积。$${\mu _{\rho ,\left\{ {0,{d_k}} \right\}}}$$上的非谱问题是研究$${L^2中正交指数的基数}\left( {{\mu _{\rho ,\left\{ {0,{d_k}} \right\}}}} \right)$$ . 在本文中，我们通过对 ρ 的值进行分类来表征该问题。