On small values of indefinite diagonal quadratic forms at integer points in at least five variables

Abstract

For anyε>0\varepsilon > 0we derive effective estimates for the size of a non-zero integral pointm∈Zd∖{0}m \in \mathbb {Z}^d \setminus \{0\}solving the Diophantine inequality|Q[m]|>ε\lvert Q[m] \rvert > \varepsilon, whereQ[m]=q1m12+…+qdmd2Q[m] = q_1 m_1^2 + \ldots + q_d m_d^2denotes a non-singular indefinite diagonal quadratic form ind≥5d \geq 5variables. In order to prove our quantitative variant of the Oppenheim conjecture, we extend an approach developed by Birch and Davenport to higher dimensions combined with a theorem of Schlickewei. 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)(r,s), to diagonal forms up to a negligible growth factor.