  We show that the Maxwell-Klein-Gordon equations in three dimensions are
globally well-posed in $H^s_x$ in the Coulomb gauge for all $s > \sqrt{3}/2
\approx 0.866$. This extends previous work of Klainerman-Machedon
\cite{kl-mac:mkg} on finite energy data $s \geq 1$, and Eardley-Moncrief
\cite{eardley} for still smoother data. We use the method of almost
conservation laws, sometimes called the "I-method", to construct an almost
conserved quantity based on the Hamiltonian, but at the regularity of $H^s_x$
rather than $H^1_x$. One then uses Strichartz, null form, and commutator
estimates to control the development of this quantity. The main technical
difficulty (compared with other applications of the method of almost
conservation laws) is at low frequencies, because of the poor control on the
$L^2_x$ norm. In an appendix, we demonstrate the equations' relative lack of
smoothing - a property that presents serious difficulties for studying rough
solutions using other known methods.
