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Simultaneous diophantine approximation with quadratic and linear forms
Let $Q$ be a nondegenerate indefinite
quadratic form on $\mathbb{R}^n$, $n\geq 3$, which is not a scalar
multiple of a rational quadratic form, and let $C_Q=\{v\in \mathbb R^n |
Q(v)=0\}$. We show that given $v_1\in C_Q$, for
almost all $v\in C_Q \setminus \mathbb R v_1$ the following holds:
for any $a\in \mathbb R$, any affine plane $P$ parallel to the
plane of $v_1$ and $v$, and $\epsilon >0$ there exist primitive
integral $n$-tuples $x$ within $\epsilon $ distance of $P$ for which
$|Q(x)-a|<\epsilon$. An analogous result is also proved for almost all
lines on $C_Q$.