Generalizations of Scott's inequality and Pick's formula to rational polygons

We prove a sharp upper bound on the number of boundary lattice points of a rational polygon in terms of its denominator and the number of interior lattice points, generalizing Scott's inequality. We then give sharp lower and upper bounds on the area in terms of the denominator, the number of interior lattice points, and the number of boundary lattice points, which can be seen as a generalization of Pick's formula. Minimizers and maximizers are described in detail. As an application, we derive bounds for the coefficients of Ehrhart quasipolymials of half-integral polygons.