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We show that there exists $\eta > 0$ such that the interval $[X, X + X^{\frac15 - \eta}]$ contains a squarefree number for all large $X$. This improves onan earlier result of Filaseta and Trifonov who showed that there is asquarefree number in $[X, X + cX^{\frac 15}\log X]$ for some $c > 0$ and alllarge $X$.  We introduce a new technique to count lattice points near curves, which weuse to bound in critical ranges the number of integers in a short intervaldivisible by a large square. This uses as an input Green and Tao's quantitativeversion of Leibman's theorem on the equidistribution of polynomial orbits innilmanifolds.

Squarefree numbers in short intervals