Lattice points in high-dimensional spheres

Abstract

LetN(x, n, α) denote the number of integer lattice points inside then-dimensional sphere of radius (an)1/2 with center at x. This numberN(x,n, α) is studied for α fixed,n → ∞, andx varying. The average value (asx varies) ofN(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 ofN (x,n, α) differ from the everage by factors exponential inn, 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.