Algorithmic decidability of the doubling property for non-Pisot β

Develop an algorithm that, given a real number β in (1, 2) that is not a Pisot number and a probability weight p = (p_1, p_2) with p_1, p_2 > 0 and p_1 + p_2 = 1, decides whether the self-similar measure μ_p on [0,1] associated with S_1(x) = x/β and S_2(x) = x/β + (1 − 1/β) is a doubling measure.

Background

For Pisot β in (1,2), the authors’ method yields an algorithm to determine whether the corresponding self-similar measure μp is doubling. They prove non-doubling for a specific family of Pisot β (those satisfying β^m = ∑{j=0}^{m-1} β^j with m ≥ 3).

In contrast, for β not Pisot, the authors explicitly state the absence of any algorithm to check the doubling property of μ_p. This highlights a computational and theoretical gap: deciding the doubling property for non-Pisot parameters remains open.