  We show that there exists $\eta > 0$ such that the interval $[X, X + X^{\frac
15 - \eta}]$ contains a squarefree number for all large $X$. This improves on
an earlier result of Filaseta and Trifonov who showed that there is a
squarefree number in $[X, X + cX^{\frac 15}\log X]$ for some $c > 0$ and all
large $X$.
  We introduce a new technique to count lattice points near curves, which we
use to bound in critical ranges the number of integers in a short interval
divisible by a large square. This uses as an input Green and Tao's quantitative
version of Leibman's theorem on the equidistribution of polynomial orbits in
nilmanifolds.
