  This article concerns the Cauchy problem for the gravity-capillary water
waves system in general dimensions. We establish local well-posedness for
initial data in $H^s$, with $s > \frac{d}{2} + 2 - \mu$, with $\mu =
\frac{3}{14}$ and $\mu = \frac37$ in the cases $d = 1$ and $d \geq 2$
respectively. This represents an improvement over the the state-of-the-art low
regularity theory in $d \geq 2$ dimensions.
