  We introduce a new non-resonant low-regularity integrator for the cubic
nonlinear Schr\"odinger equation (NLSE) allowing for long-time error estimates
which are optimal in the sense of the underlying PDE. The main idea thereby
lies in treating the zeroth mode exactly within the discretization. For
long-time error estimates, we rigorously establish the long-time error bounds
of different low-regularity integrators for the nonlinear Schr\"odinger
equation (NLSE) with small initial data characterized by a dimensionless
parameter $\varepsilon \in (0, 1]$. We begin with the low-regularity integrator
for the quadratic NLSE in which the integral is computed exactly and the
improved uniform first-order convergence in $H^r$ is proven at $O(\varepsilon
\tau)$ for solutions in $H^r$ with $r > 1/2$ up to the time $T_{\varepsilon } =
T/\varepsilon $ with fixed $T > 0$. Then, the improved uniform long-time error
bound is extended to a symmetric second-order low-regularity integrator in the
long-time regime. For the cubic NLSE, we design new non-resonant first-order
and symmetric second-order low-regularity integrators which treat the zeroth
mode exactly and rigorously carry out the error analysis up to the time
$T_{\varepsilon } = T/\varepsilon ^2$. With the help of the regularity
compensation oscillation (RCO) technique, the improved uniform error bounds are
established for the new non-resonant low-regularity schemes, which further
reduce the long-time error by a factor of $\varepsilon^2$ compared with
classical low-regularity integrators for the cubic NLSE. Numerical examples are
presented to validate the error estimates and compare with the classical
time-splitting methods in the long-time simulations.
