Lattice points in high-dimensional spheres

Let N(x, n, α) denote the number of integer lattice points inside the n-dimensional sphere of radius (an)^1/2 with center at x. This number N(x, n, α) is studied for α fixed, n → ∞, and x varying. The average value (as x varies) of N(x, n, α) is just the volume of the sphere, which is roughly of the form (2 βe, α)^n/2. it is shown that the maximal and minimal values of N (x, n, α) differ from the everage by factors exponential in n, which is in contrast to the usual lattice point problems in bounded dimensions. This lattice point problem arose separately in universal quantization and in low density subset sum problems.