Distribution of values of quadratic forms at integral points

Abstract

AbstractThe number of lattice points in d-dimensional hyperbolic or elliptic shells $$\{m : a<Q[m]<b\}$$ { m : a < Q [ m ] < b } , which are restricted to rescaled and growing domains $$r\,\Omega $$ r Ω , is approximated by the volume. An effective error bound of order $$o(r^{d-2})$$ 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 $$d \ge 9$$ d ≥ 9 to dimension $$d \ge 5$$ d ≥ 5 . They apply to wide shells when $$b-a$$ 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 $$d\ge 5$$ d ≥ 5 solving the Diophantine inequality $$|Q[m] |< \varepsilon $$ | Q [ m ] | < ε and provide error bounds comparable with those for positive forms up to powers of $$\log r$$ log r .