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(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),

Additive inhomogeneous Diophantine inequalities

“…From this point onward, we follow [13] quite closely, which we note was in turn motivated by the work of Bentkus and Götze [2]. We have the following lemma, which is similar to Lemma 3 of [13] and to Theorem 6.1 of [2].…”

“…We use the DavenportHeilbronn method, and in addition some of the recent ideas of Bentkus and Götze [2]. We also rely heavily on the methods of Davenport and Roth [10].…”

“…For if one considers equations such as (G(x)) 3 − 2 = 0, (2) where G(x) is an integral polynomial, we can see that there is good reason to restrict to the homogeneous case: the number of variables s here can be chosen as large as we like, and G(x) can even be chosen so that the equation (2) has real solutions, yet there are clearly no integral solutions of (2).…”

“…We use the Davenport-Heilbronn method, with variations based on the ideas of Bentkus and Götze [2]. We note that we need not use the ideas of Bentkus and Götze to attack the case in which H(x) is an indefinite polynomial.…”

“…(For the traditional Weyl inequality, see, for example, Lemma 2.4 of [18].) Our aim is to show, following the method of [2], that there is a function T (P ), which tends to infinity as P tends to infinity, such that we have…”