Distribution of Values of Quadratic Forms at Integral Points

TL;DR

This paper provides effective bounds for counting lattice points in hyperbolic or elliptic shells defined by quadratic forms, extending previous results to lower dimensions and indefinite forms, with applications to Diophantine inequalities.

Contribution

It introduces explicit error bounds for lattice point counts in shells for quadratic forms, improving previous non-effective results and extending to indefinite forms in lower dimensions.

Findings

Effective error bounds of order o(r^{d-2}) for lattice point counts

Extension of effective results to dimensions d ≥ 5 for positive definite forms

Explicit bounds for solutions to Diophantine inequalities for indefinite forms

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 $r\;\Omega$, 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 $d \geq 9$ 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 $\log r$.