Presburger Arithmetic with algebraic scalar multiplications

We consider Presburger arithmetic (PA) extended by scalar multiplication an algebraic irrational number $\alpha$, and call this extension $\alpha$-Presburger ($\alpha$-PA). show that the complexity of deciding sentences in $\alpha$-PA is substantially harder than PA. Indeed, when $\alpha$ quadratic $r\geq 4$, with $r$ alternating quantifier blocks at most $c\ r$ variables inequalities requires space least $K 2^{\cdot^{\cdot^{\cdot^{2^{C\ell(S)}}}}}$ (tower height $r-3$), where constants $c, K, C>0$ only depend on $\ell(S)$ length given sentence $S$. Furthermore $\exists^{6}\forall^{4}\exists^{11}$ $k$ PSPACE-hard, another constant depending on~$\alpha$. When non-quadratic, already four suffice for undecidability sentences.