Distribution of values of quadratic forms at integral points

• 成果类型: Article
• 署名作者: Buterus, P.; Goetze, F.; Hille, T.; Margulis, G.
• 署名单位: University of Gottingen; University of Bielefeld; Northwestern University; Yale University
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-021-01086-6
• 发表日期: 2022
• 页码: 857-961
• 关键词: pair correlation densities; quantitative version; small zeros; subspaces; THEOREM

摘要

The number of lattice points in d-dimensional hyperbolic or elliptic shells {m : a < Q[m] < b}, which are restricted to rescaled and growing domains r Omega, is approximated by the volume. An effective error bound of order o(r(d-2)) for this approximation is proved based on Diophantine approximation properties of the quadratic form Q. These results allow to show effective variants of previous non-effective results in the quantitative Oppenheim problem and extend known effective results in dimension d >= 9 to dimension d >= 5. They apply to wide shells when b - a is growing with r and to positive definite forms Q. For indefinite forms they provide explicit bounds (depending on the signature or Diophantine properties of Q) for the size of non-zero integral points m in dimension d >= 5 solving the Diophantine inequality vertical bar Q[m]vertical bar < epsilon and provide error bounds comparable with those for positive forms up to powers of log r.