Beurling dimension of a class of spectra of the Sierpinski-type spectral measures

It is known that the Beurling dimensions of spectra of the self-similar measure are bounded by the Hausdorff dimension of its support and the bound can be attained. The relationship is also true for the self-affine case. But we are not sure whether the upper bound can be attained, except a few particular cases with fine digit sets. In this paper, based on a subtle relationship between the self-similar set in dimension one and the projection of the candidate spectrum, we determine the exact Beurling dimension for a class of spectra of the Sierpinski-type spectral measures. It is strictly less than the Hausdorff dimension of the support for the measure, which is in stark contrast with the self-similar case.