  The $N$-jettiness observable $\mathcal{T}_N$ provides a way of describing the
leading singular behavior of the $N$-jet cross section in the $\tau
=\mathcal{T}_N/Q \to 0$ limit, where $Q$ is a hard interaction scale. We
consider subleading power corrections in the $\tau \ll 1$ expansion, and employ
soft-collinear effective theory to obtain analytic results for the dominant
$\alpha_s \tau \ln\tau$ and $\alpha_s^2 \tau\ln^3\tau$ subleading terms for
thrust in $e^+e^-$ collisions and $0$-jettiness for $q\bar q$-initiated
Drell-Yan-like processes at hadron colliders. These results can be used to
significantly improve the numerical accuracy and stability of the $N$-jettiness
subtraction technique for performing fixed-order calculations at NLO and NNLO.
They reduce the size of missing power corrections in the subtractions by an
order of magnitude. We also point out that the precise definition of
$N$-jettiness has an important impact on the size of the power corrections and
thus the numerical accuracy of the subtractions. The sometimes employed
definition of $N$-jettiness in the hadronic center-of-mass frame suffers from
power corrections that grow exponentially with rapidity, causing the power
expansion to deteriorate away from central rapidity. This degradation does not
occur for the original $N$-jettiness definition, which explicitly accounts for
the boost of the Born process relative to the frame of the hadronic collision,
and has a well-behaved power expansion throughout the entire phase space.
Integrated over rapidity, using this $N$-jettiness definition in the
subtractions yields another order of magnitude improvement compared to
employing the hadronic-frame definition.
