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 $(\alpha,\beta)\in[0,1)\times(1,\infty)$ 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.