Lattice points in regions
Ivan Niven, Herbert S. Zuckerman · Proceedings of the American Mathematical Society · 1967
1. Let S be a bounded set of points in the Euclidean plane with a unit distance defined. If a rectangular coordinate system is imposed, a certain number of points of S are lattice points, i.e. points with integer coordinates. Let m(S) be the minimum number of lattice points of S under all possible choices of the axis system, and M(S) the maximum number. For example if S is a closed disk of diameter one, then mr(S) = 0 and M(S) =2. The definitions of mr(S) and M(S) could be given in terms of a fixed rectangular coordinate system, with the set S being freely rotated and translated in the plane. It will be convenient in the proofs to use sometimes one and sometimes the other of these two formulations. Although the definitions and theorems of this paper are given for 2-dimensional Euclidean space, the generalization to higher dimensions involves no difficulties whatsoever. It is apparent that mr(S) < M(S) for any set S. R. M. Robinson suggested that the strict inequality holds for a nonempty bounded closed set, which is a more general result than we had formulated.