Post #3644455

Does anyone know a reference for the following easy theorem about estimating area by counting lattice points? Let \(J\) be a region of area \(a) bounded by a Jordan curve of length \(p\). Then: \[|a - \#(\mathbb{Z}^2\cap J)| = O(p).\] Proof: sweep a unit square around the boundary of \(J\); by Cavalieri's principle the area of the swept region \(B\) is \(\le p\sqrt 2\). Consider the Voronoi cells of the integer lattice; outside of \(B\) they are completely inside or completely outside \(J\). Therefore, \[a-p\sqrt 2\le \operatorname{area}(J\setminus B)\le \#(\mathbb{Z}^2\cap J)\le \operatorname{area}(J\cup B)\le a+p\sqrt 2.\] #ProofInAToot