Presburger Arithmetic with algebraic scalar multiplications

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