= Solution
One Archimedean form of the <Schmidt subspace theorem> is as follows. Let $L_1,\ldots,L_n$ be linearly independent <linear form>[linear forms] in $n$ variables with algebraic coefficients. For every $\varepsilon>0$, all nonzero $\mathbf x\in\mathbb Z^n$ satisfying
$$
\prod_{i=1}^n|L_i(\mathbf x)|
<\|\mathbf x\|^{-\varepsilon}
$$
belong to a finite union of proper rational <linear subspace>[linear subspaces] of $\mathbb Q^n$.
We will also use its finite-place form: if $S$ is a finite set of places containing the Archimedean ones and, for each $v\in S$, the forms $L_{1,v},\ldots,L_{n,v}$ are independent, then the integer solutions of
$$
\prod_{v\in S}\prod_{i=1}^n|L_{i,v}(\mathbf x)|_v
<H(\mathbf x)^{-\varepsilon}
$$
lie in finitely many proper rational subspaces. The absolute values are normalized so that the <product formula> holds.
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