Critical Dimensions for counting Lattice Points in Euclidean Annuli

We study the number of lattice points in R-d, d >= 2, lying inside an annulus as a function of the centre of the annulus. The average number of lattice points there equals the volume of the annulus, and we study the L-1 and L-2 norms of the remainder. We say that a dimension is critical, if these norms do not have upper and lower bounds of the same order as the radius goes to infinity. In [6], it was proved that in the case of the ball (instead of an annulus) the critical dimensions are d equivalent to 1 mod 4. We show that the behaviour of the width of an annulus as a function of the radius determines which dimensions are critical now. In particular, if the width is bounded away from zero and infinity, the critical dimensions are d equivalent to 3 mod 4; if the width goes to infinity, but slower than the radius, then all dimensions are critical, and if the width tends to zero as a power of the radius, then there are no critical dimensions.