Magnetic gap-labelling conjecture for principal solenoidal tori

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 $\Theta$. The magnetic gap-labelling conjecture states that the magnetic gap-labelling group is contained in the magnetic frequency group. This conjecture gives an explicit calculation of the possible labels of spectral gaps for magnetic Schrödinger operators via the trace on the twisted crossed-product algebra. The paper proves the conjecture for principal solenoidal tori in all dimensions, so the claim is solved.