Lattice Point Problems and Distribution of Values of Quadratic Forms

Abstract

For d-dimensional irrational ellipsoids E with d > 9 we show that the number of lattice points in rE is approximated by the volume of rE, as r tends to infinity, up to an error of order o(rd-2). The estimate refines an earlier authors' bound of order O(rd-2) which holds for arbitrary ellipsoids, and is optimal for rational ellipsoids. As an application we prove a conjecture of Davenport and Lewis that the gaps between successive values, say s 0 as s - oo, for d > 9. For comparison note that sups(n(s) - s) 0, for rational Q[x] and d > 5. As a corollary we derive Oppenheim's conjecture for indefinite irrational quadratic forms, i.e., the set Q[Zd] is dense in R, for d > 9, which was proved for d > 3 by G. Margulis [Marl] in 1986 using other methods. Finally, we provide explicit bounds for errors in terms of certain characteristics of trigonometric sums.