When does a Bernoulli convolution admit a spectrum? (Q456776)

A discrete subset \(\Lambda\subset\mathbb{C}\) is a spectrum of a probability measure \(\mu\) if \(\{\exp(-2\pi i\lambda):\lambda\in \Lambda\}\) creates an orthogonal basis for \(L^2(\mu)\). A spectral measure is a probability measure that admits a spectrum. It is proved that the distribution of NEWLINE\[NEWLINE\sum^\infty_{n=1} \varepsilon_n\rho^n/2,NEWLINE\]NEWLINE called Bernoulli convolution, where \(\varepsilon_n\in \{-1,+1\}\) are choosen independently with probability \(1/2\), is not a spectral measure for irrational contraction rates \(\rho\) and that in the case of rational \(\rho\) the Bernoulli convolution fails to be a spectral measure unless \(\rho\) is the reciprocal of an even integer.