Squarefree numbers in short intervals

Abstract

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 satisfying further restrictions, 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.

arXiv comment33 pages. Significant revisions. Exponents made explicit in elementary sections, table of symbols added. Errors corrected