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

Abstract

For any $\varepsilon > 0$ we derive effective estimates for the size of a non-zero integral point $m \in \mathbb{Z}^d \setminus \{0\}$ solving the Diophantine inequality $\lvert Q[m] \rvert < \varepsilon$, where $Q[m] = q_1 m_1^2 + \ldots + q_d m_d^2$ denotes a non-singular indefinite diagonal quadratic form in $d \geq 5$ 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.