Abstract: Let $Q$ be a non-degenerate quadratic form in $d \geq 5$ variables. For a fixed interval $[a,b]$ and a domain $\Omega$ in $\mathbb R^d$ let $E_{a,b}(r)$ be the shell $a <Q[x]<b$ restricted to the dilated domain $r \Omega$. We will discuss the problem of counting the number of integral points contained in $E_{a,b}(r)$ by approximating it to its volume. We prove an effective error bound of order $o(r^{d-2})$ based on Diophantine properties of the quadratic form.

This extends effective results obtained by Bentkus and Götze in $d \geq 9$ to $d \geq 5$. In the indefinite case this yields effective error terms in the work of Eskin, Margulis and Mozes in the case $d \geq 5$. This is based on joint work with P. Buterus, F. Götze and G. Margulis.