Coefficient functions of the Ehrhart quasi-polynomials of rational polygons

In 1976, P. R. Scott characterized the Ehrhart polynomials of convex integral polygons. We study the same question for Ehrhart polynomials and quasi-polynomials of \emph{non}-integral convex polygons. Define a \emph{pseudo-integral polygon}, or \emph{PIP}, to be a convex rational polygon whose Ehrhart quasi-polynomial is a polynomial. The numbers of lattice points on the interior and on the boundary of a PIP determine its Ehrhart polynomial. We show that, unlike the integral case, there exist PIPs with $b=1$ or $b=2$ boundary points and an arbitrary number $I \ge 1$ of interior points. However, the question of whether a PIP must satisfy Scott's inequality $b \le 2I + 7$ when $I \ge 1$ remains open. Turning to the case in which the Ehrhart quasi-polynomial has nontrivial quasi-period, we determine the possible minimal periods that the coefficient functions of the Ehrhart quasi-polynomial of a rational polygon may have.