Lattice point problems and values of quadratic forms
Götze, Friedrich
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.
SPRINGER-VERLAG
2004
info:eu-repo/semantics/article
doc-type:article
text
https://pub.uni-bielefeld.de/record/1607761
Götze F. Lattice point problems and values of quadratic forms. <em>INVENTIONES MATHEMATICAE</em>. 2004;157(1):195-226.
eng
info:eu-repo/semantics/altIdentifier/doi/10.1007/s00222-004-0366-3
info:eu-repo/semantics/altIdentifier/issn/0020-9910
info:eu-repo/semantics/altIdentifier/wos/000221850600006
info:eu-repo/semantics/closedAccess