Uniform Distribution

At primary school the first author was taught to estimate the area of a (convex) body by drawing it on a piece of graph paper, and then counting the number of (unit) squares inside. There is obviously a little ambiguity in deciding how to count the squares which straddle the boundary. Whatever the protocol, if the boundary is more-or-less smooth then the number of squares in question is proportional to the perimeter of the body, which will be small compared to the area (if the body is big enough). At secondary school the first author learnt that there are other methods to determine areas, sometimes more precise. As an undergraduate he learned that counting lattice points is often a difficult question (and that counting unit squares is “equivalent” to counting the lattice points in their bottom left-hand corner). Then, as a graduate student, he learnt that the primary school method could be turned around to provide a good tool for estimating the number of lattice points inside a convex body! In the specific case of a right-angled triangle we fix the slope −α of the hypotenuse and ask for the number of lattice points Aα(N) := #{(x, y) ∈ Z2 : x, y ≥ 0 and y + αx ≤ N}.