  We show that counts of squarefree integers up to $X$ in short intervals of
size $H$ tend to a Gaussian distribution as long as $H\rightarrow\infty$ and $H
= X^{o(1)}$. This answers a question posed by R.R. Hall in 1989. More generally
we prove a variant of Donsker's theorem, showing that these counts scale to a
fractional Brownian motion with Hurst parameter $1/4$. In fact we are able to
prove these results hold in general for collections of $B$-free integers as
long as the sieving set $B$ satisfies a very mild regularity property, for
Hurst parameter varying with the set $B$.
