On the triple jump of the set of atoms of a Boolean algebra

We prove the following result concerning the degree spectrum of the atom relation on a computable Boolean algebra. Let $\mathcal{C}$ be a computable Boolean algebra with infinitely many atoms and $\mathbf{a}$ be the Turing degree of the atom relation of $\mathcal{C}$. If $\mathbf{d}$ is a c.e. degree such that $\mathbf{a}^{\prime\prime\prime}\leq_T\mathbf{d}^{\prime\prime\prime}$, then there is a computable copy of $\mathcal{C}$ where the atom relation has degree $\mathbf{d}$. In particular, for every $\mathrm{high}_3$ c.e. degree $\mathbf{d}$, any computable Boolean algebra with infinitely many atoms has a computable copy where the atom relation has degree $\mathbf{d}$.