Hausdorff dimension of the parameters for (α,β)-transformations with the specification property

In this paper we consider the specification property for (α,β)-shifts. When α=0, Schmeling shows that the set of β>1 for which the β-shift has the specification property has the Lebesgue measure zero but has the full Hausdorff dimension{Schmeling}. So it is natural to ask what happens when α>0. Buzzi shows that for fixed α the set of β >1 for which the (α,β)-shift has the specification property has Lebesgue measure zero. Hence we consider the Hausdorff dimension of the parameter space of (α,β)-shifts.