Effective density of values of indefinite ternary inhomogeneous quadratic forms

Abstract

Given an inhomogeneous quadratic form $Q_{\xi }(v)=Q(v+\xi )$ with $Q$ an indefinite $\mathbb{Q}$-isotropic rational ternary form and $\xi \in {\mathbb{R}}^3$ irrational, we prove an effective lower bound for the number of integer vectors $v\in {\mathbb{Z}}^n$ with $\Vert v\Vert \le T$ such that $|Q_{\xi }(v)-t|<\delta$ that is valid for any $t\in \mathbb{R}$ and all $\delta \ge T^{-\nu }$, with $\nu >0$ depending explicitly on the Diophantine properties of $\xi$. In particular, for $\xi$ with algebraic entries we can take any $\nu <\frac{1}{8}$.

8 pages