Spectrality of non-homogeneous self-similar measures

Determine, for contraction ratios \(\rho_1\neq\rho_2\) and weight \(p\in(0,1)\), exactly when the non-homogeneous self-similar measure \(\mu_{\rho_1,\rho_2}^p\) generated by \(f_{1,\rho_1}(x)=\rho_1x\) and \(f_{2,\rho_2}(x)=\rho_2(x+2)\) is a spectral measure.

Background

The paper studies self-similar measures whose two contraction ratios differ. Unlike classical Bernoulli convolutions, these measures generally lack an infinite convolution representation by finitely supported measures, which prevents the direct application of standard methods for constructing exponential orthonormal bases.

The authors identify the general spectrality problem as a long-standing question and then specialize their analysis to the golden-mean case $\rho_1=1/2$, $\rho_2=1/4$, and $p=(1/2)^s$. The paper establishes structural and Fourier-analytic properties of that particular measure but does not determine its spectrality.