Spectrality of the golden-mean self-similar measure

Determine whether the golden-mean self-similar measure with contraction ratios \(\rho_1=1/2\), \(\rho_2=1/4\), and weight \(p=(1/2)^s\), where \((1/2)^s+(1/4)^s=1\), admits an exponential orthonormal basis in its associated \(L^2\)-space.

Background

The golden-mean self-similar measure is the paper’s principal example of a non-homogeneous self-similar measure. Its Fourier transform satisfies a non-homogeneous functional equation, and the measure cannot be represented as an infinite convolution of Bernoulli measures.

The paper proves that the Fourier transform has no zeros on a small interval and reports numerical evidence for the absence of real zeros on a much larger interval. These results strongly suggest non-spectrality, but they do not resolve whether the measure possesses an exponential orthonormal basis.