  We investigate a filtered Lie-Trotter splitting scheme for the ``good"
Boussinesq equation and derive an error estimate for initial data with very low
regularity. Through the use of discrete Bourgain spaces, our analysis extends
to initial data in $H^{s}$ for $0<s\leq 2$, overcoming the constraint of
$s>1/2$ imposed by the bilinear estimate in smooth Sobolev spaces. We establish
convergence rates of order $\tau^{s/2}$ in $L^2$ for such levels of regularity.
Our analytical findings are supported by numerical experiments.
