Distribution of values of quadratic forms at integral points

Abstract

Abstract The number of lattice points in d -dimensional hyperbolic or elliptic shells 1517135\{m : a<Q[m]<b\}1517135 { m : a < Q [ m ] < b } , which are restricted to rescaled and growing domains 1517135r\,\Omega 1517135 r Ω , is approximated by the volume. An effective error bound of order 1517135o(r^{d-2})1517135 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 1517135d \ge 91517135 d ≥ 9 to dimension 1517135d \ge 51517135 d ≥ 5 . They apply to wide shells when 1517135b-a1517135 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 1517135d\ge 51517135 d ≥ 5 solving the Diophantine inequality 1517135