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 ge 9 to dimension d ge 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 dge 5 solving the Diophantine inequality |Q[m] |< varepsilon and provide error bounds comparable with those for positive forms up to powers of log r.
Highlights
Points m in dimension d ≥ 5 solving the Diophantine inequality |Q[m]| < ε and provide error bounds comparable with those for positive forms up to powers of log r
Raghunathan, between the Oppenheim conjecture and questions concerning closures in SL(3, R)/SL(3, Z) of orbits of certain subgroups of SL(3, R). It is based on the study of minimal invariant sets and the limits of orbits of sequences of points tending to a minimal invariant set
We would like to study the distribution of values of Q at integer points, often referred to as “quantitative Oppenheim conjecture” with an emphasis on establishing effective error bounds for the approximation of the number of lattice points restricted to growing domains
Summary
The situation with asymptotics and upper bounds is more subtle It was proved in [23] that if Q is an irrational indefinite quadratic form of signature ( p, q) with p + q = d, p ≥ 3 and q ≥ 1, for any a < b lim volZ (Ea,b ∩ r ) = 1 r→∞ vol (Ea,b ∩ r ). While the asymptotics as in (1.4) do not hold in the case of signatures (2, 1) and (2, 2), one can show (see [23]) that in these cases there is an upper bound of the form r d−2 log r. Remark 1.2 The proofs of the above mentioned results use such notions as a minimal invariant set (in the case of the Oppenheim conjecture) and an ergodic invariant measure These notions do not have in general effective analogs. Because of that it is very difficult to get ‘good’ estimates for the size of the smallest non-trivial integral solution of the inequality |Q[m]| < ε and ‘good’ error terms in the quantitative Oppenheim conjecture by applying dynamical and ergodic methods
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