  We compute the leading logarithmic power corrections at
next-to-next-to-next-to-leading order for $0$-jettiness subtractions for color
singlet production. We discuss how to disentangle these power corrections from
those arising from the presence of fiducial and isolation cuts by using
Projection-to-Born improved slicing. We present the results for Drell-Yan and
Higgs production in gluon fusion differential in both the invariant mass and
rapidity of the color singlet. Our results include all the channels
contributing at leading logarithmic order for these processes, including the
off-diagonal channels that receive contributions from soft quark emission. We
study the numerical impact of the power corrections for Drell-Yan and Higgs
production and find it to become negligible compared to the size of the N$^3$LO
corrections only below $\tau_\text{cut} \sim 10^{-5}$. We estimate that in a
fully differential calculation at N$^3$LO combining the Projection-to-Born
improved slicing method and our results for the leading logarithmic power
corrections may allow for keeping the slicing uncertainties under control
already with $\tau_\text{cut} \lesssim 10^{-3}$, marking a significant
improvement in efficiency for these methods. These results constitute a crucial
ingredient for fully differential N$^3$LO calculations based on the
$N$-jettiness subtraction scheme.
