Hausdorff dimension of specification for the $(\alpha ,\beta )$-shifts

Specification is an important concept in dynamical systems introduced by Bowen. Schmeling proved that the set of $\beta >1$ such that the corresponding $\beta$-shift has specification is of Hausdorff dimension $1$. Hu et al. proved that the set of $\beta >1$ such that the corresponding $(-\beta )$-shift has specification is of Hausdorff dimension $1$. We show that the set of $(\alpha ,\beta )\in [0,1)\times (1,\mathrm{\infty })$ such that the corresponding $(\alpha ,\beta )$-shift has specification is of Hausdorff dimension $2$. A new difficulty is a simultaneous control of two critical symbol sequences that determine the ambient shift space. We achieve this by taking intersections of two thick Cantor sets in parameter space.