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

Paul Buterus, Friedrich Götze

Published 2018-10-28Version 1

For any $\epsilon >0$ we derive an effective estimate for a solution of $|Q[m]| < \epsilon$ in non-zero integral points $m \in \mathbb Z^d \setminus \{0\}$ in terms of the signature $(r,s)$ and the largest eigenvalue, where $Q[x] = \sum_{i=1}^d \lambda_i x_i^2$ is a non-singular indefinite diagonal quadratic form of dimension $d \geq 5$. In order to prove our result, we extend an approach of Birch and Davenport(1958b) to higher dimensions combined with a theorem of Schlickewei (1985) on small zeros of integral quadratic forms.