Left panel: redshift evolution of the gas fraction f_gas for our sample of dusty AGN. The targets are colour-coded according to the distance from the main sequence of SFGs (see colour bar in the bottom right corner). For BzK8608 and CDF153 we also show (with open symbols) the upper limits we would obtain by assuming α_CO = 0.8 instead of 3.6 (see Sect. 2). The predicted evolutionary trend of MS galaxies of M_star = 10¹¹ M_⊙ from the 2-SFM model is shown with a solid curve (shaded area: 1σ scatter around average evolutionary trend line). We also show the predicted trends for MS galaxies of M_star = 2 × 10¹⁰ M_⊙ and M_star = 5 × 10¹¹, as labelled in the figure. The error bar in the bottom left corner of the plot shows the representative 1σ error for the data points (we assume a 0.2 dex uncertainty for M_star estimates; see Table C.2). Right panel: redshift evolution of the normalised gas fraction μ_gas for our sample of dusty AGN. The predicted evolutionary trend of “gas depleted” MS galaxies of M_star = 10¹¹ M_⊙ is shown with the black curve (the shaded area indicates 1σ scatter around the average trend). This trend has been obtained in the framework of the 2-SFM model (Sargent et al. 2014) by considering the depletion of gas due to AGN-driven outflows (M_out = 5 × 10¹⁰ M_⊙). We also display the predicted trends for M_out = 10¹⁰ and 10¹¹ M_⊙. The blue symbols refer instead to the predicted μ_gas of high-z unobscured QSOs from Valiante et al. (2014). The star symbols are associated with X-ray detected AGN and are identical to those used in Fig. 3. Essentially all obscured AGN hosts show a gas fraction deficit compared to normal galaxies, with CT QSOs (large star symbols) displaying some of the strongest deficits.