L O ] 9 M ay 2 01 8 PRESBURGER ARITHMETIC WITH ALGEBRAIC SCALAR MULTIPLICATIONS

We study complexity of integer sentences in Sα = (R, <,+,Z, x 7→ αx), which is known to be decidable for quadratic α, and undecidable for non-quadratic irrationals. When α is quadratic and the sentence has r alternating quantifier blocks, we prove both lower and upper bounds as towers of height (r− 3) and r, respectively. We also show that for α non-quadratic, already r = 4 alternating quantifier blocks suffice for undecidability.