Lattice point problems and values of quadratic forms

For d-dimensional ellipsoids E with dgreater than or equal to5 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)) for general ellipsoids and up to an error of order o(r(d-2)) for irrational ones. The estimate refines earlier bounds of the same order for dimensions dgreater than or equal to9. As an application a conjecture of Davenport and Lewis about the shrinking of gaps between large consecutive values of Q[m],mis an element ofZ(d) of positive definite irrational quadratic forms Q of dimension dgreater than or equal to5 is proved. Finally, we provide explicit bounds for errors in terms of certain Minkowski minima of convex bodies related to these quadratic forms.