The Bass and topological stable ranks for algebras of almost periodic functions on the real line

Let $\mathrm{\Lambda }$ be a sub-semigroup of the reals. We show that the Bass and topological stable ranks of the algebras ${\rm AP}_\Lambda={f\in {\rm AP}: \sigma(f)\subseteq \Lambda}$ of almost periodic functions on the real line and with Bohr spectrum in $\mathrm{\Lambda }$ are infinite whenever the algebraic dimension of the $\mathbb{Q}$-vector space generated by $\mathrm{\Lambda }$ is infinite. This extends Su'arez’s result for ${\mathrm{A}\mathrm{P}}_{\mathbb{R}}=\mathrm{A}\mathrm{P}$. Also considered are general subalgebras of AP.