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Extremal property of some surfaces in n-dimensional Euclidean space

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Abstract

A surface Γ=(f 1(X1,..., xm),...,f n(x1,..., xm)) 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.

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Authors and Affiliations

  1. Institute of Mathematics, Academy of Sciences of the Belorussian SSR, USSR

    V. I. Bernik & É. I. Kovalevskaya

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  1. V. I. Bernik
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  2. É. I. Kovalevskaya
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Translated from Matematicheskie Zametki, Vol. 15, No. 2, pp. 247–254, February, 1974.

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Bernik, V.I., Kovalevskaya, É.I. Extremal property of some surfaces in n-dimensional Euclidean space. Mathematical Notes of the Academy of Sciences of the USSR 15, 140–144 (1974). https://doi.org/10.1007/BF02102395

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  • DOI: https://doi.org/10.1007/BF02102395

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