Joint PU/IAS Number Theory

We count squarefree numbers in short intervals [X, X+H] for H > X^{1/5 - $\delta$}, where $\delta$ > 0 is some absolute constant. This improves on the exponent 1/5 shown by Filaseta and Trifonov in 1992. 

In improving bounds on the number of integers in a short interval divisible by a large square, we introduce a technique to count lattice points satisfying certain conditions near curves. This requires as an input Green and Tao’s quantitative version of Leibman’s theorem on the equidistribution of polynomial orbits in nilmanifolds.