Effective density of values of indefinite ternary inhomogeneous quadratic forms

Abstract

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