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

Abstract

For any ε > 0 \varepsilon > 0 we derive effective estimates for the size of a non-zero integral point m ∈ Z d ∖ { 0 } m \in \mathbb {Z}^d \setminus \{0\} solving the Diophantine inequality | Q [ m ] | > ε \lvert Q[m] \rvert > \varepsilon , where Q [ m ] = q 1 m 1 2 + … + q d m d 2 Q[m] = q_1 m_1^2 + \ldots + q_d m_d^2 denotes a non-singular indefinite diagonal quadratic form in d ≥ 5 d \geq 5 variables. 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.