Non-spectral problem on infinite Bernoulli convolution (Q2043702)

The article is devoted to the convolution of measures on \([0, \infty)\) with support in a countable set. Particularly, the infinite Bernoulli convolution \(\mu_{\rho, \{ 0, d_k \}} \) is considered as a family of Dirac measures. For them orthogonal sets and maximal orthogonal sets are studied. They correspond to orthonormal families (or maximal orthonormal families) in \(L^2(\mu_{\rho, \{ 0, d_k \}}, \mathbb{R})\). Conditions on \(\rho \) are found for which such sets exist. Examples are provided.