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 $r\;Ω$, 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 $d \geq 9$ to dimension $d \geq 5$. 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 $d\geq 5$ solving the Diophantine inequality $|Q[m]| < \varepsilon$ and provide error bounds comparable with those for positive forms up to powers of $\log r$.