@article{1607761,
abstract = {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.},
author = {GĂ¶tze, Friedrich},
issn = {0020-9910},
journal = {INVENTIONES MATHEMATICAE},
number = {1},
pages = {195--226},
publisher = {SPRINGER-VERLAG},
title = {{Lattice point problems and values of quadratic forms}},
doi = {10.1007/s00222-004-0366-3},
volume = {157},
year = {2004},
}