PUB - Publikationen an der Universität Bielefeld

For d-dimensional irrational ellipsoids E with d greater than or equal to 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(r(d-2)). The estimate refines an earlier authors' bound of order o(r(d-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 < n(s), s, n(s) E Q[Zd], of a positive definite irrational quadratic form Q[x], x is an element of R-d, are shrinking, i.e., that n(s)-s -> 0 as s -> infinity, for d greater than or equal to 9. For comparison note that sup,(n(s) - s) < infinity and inf(s)(n(s)-s) > 0, for rational Q[z] and d greater than or equal to 5. As a corollary we derive Oppenheim's conjecture for indefinite irrational quadratic forms, i.e., the set Q[Z(d)] is dense in R, for d greater than or equal to 9, which was proved for d greater than or equal to 3 by G. Margulis [Mar1] in 1986 using other methods. Finally, we provide explicit bounds for errors in terms of certain characteristics of trigonometric sums.