Lattice Points in High-Dimensional Spheres

A general principle, dating back to Gauss, that is widely used in estimating the number of integer lattice points in nice sets S in R is that this number equals the volume of S with a small error term [4,5,13]. This approach is very useful, and can be proved to be rigorous, for example, if one considers the number of lattice points in sets rT, where the dimension n is fixed, T is a given nice set, and r → ∞. However, we will show below that this principle fails completely if one considers the dimension n → ∞, and the sets S to be spheres of radii proportional to n 1/2 . That this general principle cannot be justified rigorously in this case is not surprising, since (i) the surface area of the sphere of radius (αn)1/2 is larger than the volume by a factor of (n /α)1/2 , and (ii) the diameter of the unit cube is comparable to the diameter of the sphere. However, it is rather interesting that it is not just the proof, but the principle itself, that fails.