  We present a new proof for the 1D area law for frustration-free systems with
a constant gap, which exponentially improves the entropy bound in Hastings' 1D
area law, and which is tight to within a polynomial factor. For particles of
dimension $d$, spectral gap $\epsilon>0$ and interaction strength of at most
$J$, our entropy bound is $S_{1D}\le \orderof{1}X^3\log^8 X$ where
$X\EqDef(J\log d)/\epsilon$. Our proof is completely combinatorial, combining
the detectability lemma with basic tools from approximation theory.
Incorporating locality into the proof when applied to the 2D case gives an
entanglement bound that is at the cusp of being non-trivial in the sense that
any further improvement would yield a sub-volume law.
