Lattice point problems and values of quadratic forms

Abstract

For d-dimensional ellipsoids E with d≥5 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 \(\mathcal{O}(r^{d-2})\) for general ellipsoids and up to an error of order o(rd-2) for irrational ones. The estimate refines earlier bounds of the same order for dimensions d≥9. As an application a conjecture of Davenport and Lewis about the shrinking of gaps between large consecutive values of Q[m],m∈ℤ d of positive definite irrational quadratic forms Q of dimension d≥5 is proved. Finally, we provide explicit bounds for errors in terms of certain Minkowski minima of convex bodies related to these quadratic forms.