Inferring the Distances of Fast Radio Bursts Through Associated 21-cm Absorption

The distances of Fast Radio Burst (FRB) sources are currently unknown. We show that the 21-cm absorption line of hydrogen can be used to infer the redshifts of FRB sources, and determine whether they are Galactic or extragalactic. We calculate a probability of $\sim 10\%$ for the host galaxy of an FRB to exhibit a 21-cm absorption feature of equivalent width $\gtrsim 10 ~\mathrm{km}~\mathrm{s}^{-1}$. Arecibo, along with several future radio observatories, should be capable of detecting such associated 21-cm absorption signals for strong bursts of $\gtrsim {\rm several}~\mathrm{Jy}$ peak flux densities.