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 nonzero 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 nonsingular 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.
 Publication:

arXiv eprints
 Pub Date:
 October 2018
 arXiv:
 arXiv:1810.11898
 Bibcode:
 2018arXiv181011898B
 Keywords:

 Mathematics  Number Theory;
 11D75;
 11J25
 EPrint:
 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