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