Extremal property of some surfaces in n-dimensional Euclidean space

A surface Γ=(f ₁(X₁,..., x_m),...,f _n(x₁,..., x_m)) is said to be extremal if for almost all points of Γ the inequality $$parallel a_1 f_1 (x_1 , ldots ,x_m ) + ldots + a_n f_n (x_1 , ldots ,x_m )parallel< H^{ - n - varepsilon } ,$$ , where H=max(¦a _i¦) (i=1, 2, ..., n), has only a finite number of solutions in the integersa ₁, ...,a _n. In this note we prove, for a specific relationship between m and n and a functional condition on the functionsf ₁, ...,f _n, the extremality of a class of surfaces in n-dimensional Euclidean space.