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

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