Lattice point problems and values of quadratic forms

• 成果类型: Article
• 署名作者: Götze, F
• 署名单位: University of Bielefeld
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-004-0366-3
• 发表日期: 2004
• 页码: 195-226
• 关键词: large convex-bodies; multidimensional ellipsoids; grid points; POLYNOMIALS

摘要

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.