Sharp density discrepancy for cut and project sets an approach via lattice point counting

Cut and project sets are obtained by taking an irrational slice of a lattice and projecting it to a lower dimensional subspace. We seek to quantify fluctuations from the asymptotic mean for point counts. We obtain uniform upper bounds on the discrepancy depending on the diophantine properties of the lattice. In an appendix, Michael Björklund and Tobias Hartnick obtain lower bounds on the L² 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. This also allows us to find sharp bounds for the number of lattice points in thin slabs. Using a 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; our lower bounds indicate that this is the largest class of cut and project sets for which those criteria can apply.