Characterize continuous orbit equivalence for adding-type p-adic odometers and their constant speedups

Determine elegant necessary and sufficient conditions, expressed in terms of vectors \(\mathbf{z},\mathbf{w}\in\mathbb{Z}_p^d\), for continuous orbit equivalence between the free \(\mathbb{Z}^d\)-odometers \((\mathbb{Z}_p,T_{\mathbf{z}})\) and \((\mathbb{Z}_p,T_{\mathbf{w}})\), and determine such conditions when \(T_{\mathbf{w}}\) is additionally a free minimal constant speedup of \(T_{\mathbf{z}}\).

Background

The paper proves a necessary condition for continuous orbit equivalence between free minimal adding-type $p$-adic $\mathbb{Z}^d$-odometers: there must exist $\lambda\in\mathbb{Q}_p^\times$ and $\mathbf{P}\in\mathrm{GL}_d(\mathbb{Q})$ with determinant $\pm1$ such that $\mathbf{w}=\lambda\mathbf{P}\mathbf{z}$. It then shows that this condition is not sufficient, including for free minimal constant speedups, by constructing an example satisfying the condition but failing to be continuously orbit equivalent.

The authors identify the existence of an additional, likely algebraic, obstruction and explicitly leave open the task of finding a complete characterization. The question asks both for a criterion for arbitrary pairs of free adding-type odometers and for the more specialized case in which one odometer arises as a free minimal constant speedup of the other.