Induced extremal surfaces

Under general assumptions on the functions $f_1(x),\dots,f_n(x)$ and $\varphi_1(y_1,\dots,y_k),\dots,\varphi_m(y_1,\dots,y_k)$ it is proved that the inequality
$$ \|a_1f_1+\dots+a_nf_n+a_{n+1}\varphi_1+\dots+a_{n+m}\varphi_m\|<H^{-(m+n)-\varepsilon}, $$
where $\|\alpha\|$ is the distance from $\alpha$ to the nearest integer and $H=\max|a_i|$, $i=1,\dots,n+m$, has only a finite number of solutions in integers $a_1,\dots,a_{n+m}$ for almost all $(x,y_1,\dots,y_k)\in R^{k+1}$. This establishes the extremality of the surface $(f_1,\dots,f_n,\varphi_1,\dots,\varphi_m)$.
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