PUB - Publikationen an der Universität Bielefeld

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.