On integer points in polygons (Q5915363)

The author considers the distribution of lattice points in polygons. Let \(P\) be a polygon in the plane, \(P+x\) be its shift by the vector \(x\), let \(tP\) be its image under the dilation with coefficient \(t\) with respect to the origin, and let \(S(P)\) be its area. Further let \[ R(P)= \sup_{x\in \mathbb{R}^2} |\text{card} ((P+x)\cap \mathbb{Z}^2)|- S(P). \] If the slope coefficients of the edges of a polygon are irrational, the estimation \(R(tP)= o(t)\) is well-known. The author now shows that the estimation \(R(t_\nu P)= O(t_\nu^{1/2})\) holds for some sequences \(t_\nu\to \infty\) if the edges of the polygon are rationally dependent in a sense. The proof is based on a two-dimensional Dirichlet theorem.