The magnetic gap-labelling conjecture for minimal actions

Let $\Sigma$ be a Cantor set with a minimal action of ${\mathbb Z}^p$ that preserves a Borel probability measure $\mu$. Let $\sigma$ be the multiplier on ${\mathbb Z}^p$ associated to a skew-symmetric $(p\times p)$ matrix $\Theta$. For each even-cardinality subset $I$ of $\{1,\ldots,p\}$, let $\Theta_I$ be the corresponding submatrix, let $\operatorname{Pf}(\Theta_I)$ denote its Pfaffian, and let $\mathbb Z_I[\mu]$ denote the associated frequency module.

Magnetic gap-labelling conjecture. The magnetic gap-labelling group is contained in the subgroup of $\mathbb R$ given by

$$\sum_{0\leq |I|\leq p} \operatorname{Pf}(\Theta_I)\mathbb Z_I[\mu],$$

where $\operatorname{Pf}(\Theta_\emptyset)=1$.

This conjecture proposes an explicit upper bound for the possible values of the integrated density of states in spectral gaps of magnetic operators arising from the twisted crossed product. Equality is not automatic under minimality alone and motivates the stronger conjecture for strongly minimal actions.