Non-Spectral Problem on Infinite Bernoulli Convolution

Abstract

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.