Distribution of values of quadratic forms at integral points

P. Buterus, F. Götze*, T. Hille, G. Margulis

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

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 rΩ, is approximated by the volume. An effective error bound of order o(rd-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 | Q[m] | < ε and provide error bounds comparable with those for positive forms up to powers of log r.

Original languageEnglish (US)
Pages (from-to)857-961
Number of pages105
JournalInventiones Mathematicae
Volume227
Issue number3
DOIs
StatePublished - Mar 2022

ASJC Scopus subject areas

  • Mathematics(all)

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