  We prove local well-posedness for the gravity water waves equations without
surface tension, with initial velocity field in $H^s$, $s > \frac{d}{2} + 1 -
\mu$, where $\mu = \frac{1}{10}$ in the case $d = 1$ and $\mu = \frac{1}{5}$ in
the case $d \geq 2$, extending previous results of Alazard-Burq-Zuily. The
improvement primarily arises in two areas. First, we perform an improved
analysis of the regularity of the change of variables from Eulerian to
Lagrangian coordinates. Second, we perform a time-interval length optimization
of the localized Strichartz estimates.
