Lattice point problems and distribution of values of quadratic forms

Abstract

For $d$-dimensional irrational ellipsoids $E$ with $d\ge 9$ 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 estimates refines an earlier authors’ bound of order $\mathcal{O}(r^{d-2})$ 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 $s\lt n(s)$, $s,n(s)\in Q[\mathbb{Z}^d]$, of a positive definite irrational quadratic form $Q[x]$, $x\in \mathbb{R}^d$, are shrinking, i.e., that $n(s)-s\to 0$ as $s\to \infty$, for $d\ge 9$. For comparison note that $\sup_s(n(s)-s)< \infty$ and $\inf_s(n(s)-s)>0$, for rational $Q[x]$ and $d\ge 5$. As a corollary we derive Oppenheim’s conjecture for indefinite irrational quadratic forms, i.e., the set $Q[\mathbb{Z}^d]$ is dense in $\mathbb{R}$, for $d\ge 9$, which was proved for $d\ge 3$ by G. Margulis [Mar1] in 1986 using other methods. Finally, we provide explicit bounds for errors in terms of certain characteristics of trigonometric sums.

DOI
https://doi.org/10.2307/121060

Authors

V. Bentkus

F. Götze