  We investigate the possibility of performing cosmological studies in the
redshift range $2.5<z<5$ through suitable extensions of existing and upcoming
radio-telescopes like CHIME, HIRAX and FAST. We use the Fisher matrix technique
to forecast the bounds that those instruments can place on the growth rate, the
BAO distance scale parameters, the sum of the neutrino masses and the number of
relativistic degrees of freedom at decoupling, $N_{\rm eff}$. We point out that
quantities that depend on the amplitude of the 21cm power spectrum, like
$f\sigma_8$, are completely degenerate with $\Omega_{\rm HI}$ and $b_{\rm HI}$,
and propose several strategies to independently constraint them through
cross-correlations with other probes. Assuming $5\%$ priors on $\Omega_{\rm
HI}$ and $b_{\rm HI}$, $k_{\rm max}=0.2~h{\rm Mpc}^{-1}$ and the primary beam
wedge, we find that a HIRAX extension can constrain, within bins of $\Delta
z=0.1$: 1) the value of $f\sigma_8$ at $\simeq4\%$, 2) the value of $D_A$ and
$H$ at $\simeq1\%$. In combination with data from Euclid-like galaxy surveys
and CMB S4, the sum of the neutrino masses can be constrained with an error
equal to $23$ meV ($1\sigma$), while $N_{\rm eff}$ can be constrained within
0.02 ($1\sigma$). We derive similar constraints for the extensions of the other
instruments. We study in detail the dependence of our results on the
instrument, amplitude of the HI bias, the foreground wedge coverage, the
nonlinear scale used in the analysis, uncertainties in the theoretical modeling
and the priors on $b_{\rm HI}$ and $\Omega_{\rm HI}$. We conclude that 21cm
intensity mapping surveys operating in this redshift range can provide
extremely competitive constraints on key cosmological parameters.
