Distribution of values of quadratic forms at integral points

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.