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Values of Special Indefinite Quadratic Forms

Guido Elsner
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Abstract

For special $d$-dimensional hyperbolic shells $E$ with $ d\geq 5$ we show that the number of lattice points in $E$ intersected with a $d$-dimensional cube $C_r$ of edge length $r$, can be approximated by the volume of $E\cap C_r$, as $r$ tends to infinity, up to an error of order ${\mathcal O}(r^{d-2})$. We generalize results and techniques, used by F. Götze (2004), to a large class of {\em indefinite} quadratic forms and we provide explicit bounds for the errors in terms of certain Minkowski minima related to these quadratic forms. Furthermore, we obtain, as in the positive definite case, a result for multivariate diophantine approximation and for the maximal gap between values of such indefinite forms.

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Guido Elsner. 2007-03-01. Values of Special Indefinite Quadratic Forms. https://arxiv.org/abs/math/0703029

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