Small Values of Indefinite Diagonal Quadratic Forms at Integer Points in at least five Variables

Abstract

For any $\epsilon >0$ we derive effective estimates for the size of a non-zero integral point $m\in {\mathbb{Z}}^d\setminus \{0\}$ solving the Diophantine inequality $|Q[m]|<\epsilon$, where $Q[m]=q_1{m}_1^2+\dots +q_d{m}_d^2$ denotes a non-singular indefinite diagonal quadratic form in $d\ge 5$ variables. In order to prove our quantitative variant of the Oppenheim conjecture, we extend an approach developed by Birch and Davenport [BD58b] to higher dimensions combined with a theorem of Schlickewei [Sch85]. The result obtained is an optimal extension of Schlickewei's result, giving bounds on small zeros of integral quadratic forms depending on the signature $(r,s)$, to diagonal forms up to a negligible growth factor.

This revised version removed a number of typos, improved the exposition of proofs with an appendix containing more details on results due to Schlickewei and the construction of optimal smoothing kernels