  Cut and project sets are obtained by taking an irrational slice of a lattice
and projecting it to a lower dimensional subspace, and are fully characterised
by the shape of the slice (window) and the choice of the lattice. In this
context we seek to quantify fluctuations from the asymptotics for point counts.
We obtain uniform upper bounds on the discrepancy depending on the diophantine
properties of the lattice as well as universal lower bounds on the average of
the discrepancy. In an appendix, Michael Bj\"orklund and Tobias Hartnick obtain
lower bounds on the $L^2$-norm of the discrepancy also depending on the
diophantine class; these lower bounds match our uniform upper bounds and both
are therefore sharp. Using the sufficient criteria of Burago--Kleiner and
Aliste-Prieto--Coronel--Gambaudo we find an explicit full-measure class of cut
and project sets that are biLipschitz equivalent to lattices; the lower bounds
on the variance indicate that this is the largest class of cut and project sets
for which those sufficient criteria can apply.
