Squarefree numbers in short intervals

We show that there exists $\eta >0$ such that the interval $[X,X+X^{\frac{1}{5}-\eta }]$ 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 $[X,X+c{X}^{\frac{1}{5}}\log X]$ 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.