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It is well known that, given a $2d$ purely magnetic Landau Hamiltonian with aconstant magnetic field $b$ which generates a magnetic flux $\varphi$ per unitarea, then any spectral island $\sigma_b$ consisting of $M$ infinitelydegenerate Landau levels carries an integrated density of states$\mathcal{I}_b=M \varphi$. Wannier later discovered a similar Diophantinerelation expressing the integrated density of states of a gapped group of bandsof the Hofstadter Hamiltonian as a linear function of the magnetic field fluxwith integer slope.  We extend this result to a gap labelling theorem for any $2d$ Bloch-Landauoperator $H_b$ which also has a bounded $\mathbb{Z}^2$-periodic electricpotential. Assume that $H_b$ has a spectral island $\sigma_b$ which remainsisolated from the rest of the spectrum as long as $\varphi$ lies in a compactinterval $[\varphi_1,\varphi_2]$. Then $\mathcal{I}_b=c_0+c_1\varphi$ on suchintervals, where the constant $c_0\in \mathbb{Q}$ while $c_1\in \mathbb{Z}$.The integer $c_1$ is the Chern marker of the spectral projection onto thespectral island $\sigma_b$. This result also implies that the Fermi projectionon $\sigma_b$, albeit continuous in $b$ in the strong topology, is nowherecontinuous in the norm topology if either $c_1\ne0$ or $c_1=0$ and $\varphi$ isrational.  Our proofs, otherwise elementary, do not use non-commutative geometry but arebased on gauge covariant magnetic perturbation theory which we briefly reviewfor the sake of the reader. Moreover, our method allows us to extend theanalysis to certain non-covariant systems having slowly varying magneticfields.

Beyond Diophantine Wannier diagrams: Gap labelling for Bloch-Landau Hamiltonians