Resumen de Lattice point problems and distribution of values of quadratic forms
Friedrich Götze, V. Bentkus
For d-dimensional irrational ellipsoids E with d ¡Ý 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 < n(s), s, n(s) ¡Ê Q[Zd], of a positive definite irrational quadratic form Q[x], x ¡Ê Rd, are shrinking, i.e., that n(s).s ¡ú 0 as s ¡ú ¡Þ, for d ¡Ý 9. For comparison note that sups(n(s) . s) < ¡Þand infs(n(s).s) > 0, for rational Q[x] and d ¡Ý 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 ¡Ý 9, which was proved for d ¡Ý 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.
