Critical Dimensions for counting Lattice Points in Euclidean Annuli

Department of Mathematics, University College London, Gower St., London, WC1E 6BT, UK

¹ Corresponding author. E-mail: leonid@math.ucl.ac.uk

Abstract

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 L₁ and L₂ 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 [Duke Math. J., 107 (2001), No. 2, 209–238], it was proved that in the case of the ball (instead of an annulus) the critical dimensions are d ≡ 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 ≡ 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.