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 Mat. Zametki: Year: Volume: Issue: Page: Find

 Mat. Zametki, 1978, Volume 23, Issue 2, Pages 177–181 (Mi mz8131)

A geometric property of extremal surfaces

É. I. Kovalevskaya

Institute of Mathematics, Academy of Sciences Byelorussian SSR

Abstract: Let the surface $\Gamma\in R^3$ be defined by the equation $z=f(x,y)$, where $f(x,y)$ is a function 3 times continuously differentiable in $R^2$. It is proved that if the total (Gaussian) curvature of the surface $\Gamma$ is nonzero almost everywhere on $\Gamma$ (in the sense of Lebesgue measure in $R^2$), then $\Gamma$ is extremal, i.e., for almost all $(x,y)\in R^2$ the inequality
$$\max(\|qx\|,\|qy\|,\|qf(x,y)\|)>q^{-1/3-\varepsilon},$$
holds for all integral $q\ge q_0(f)$, where $\|x\|$ is the distance from the real number $x$ to the nearest integer and $\varepsilon>0$ is arbitrarily small.

Full text: PDF file (370 kB)

English version:
Mathematical Notes, 1978, 23:2, 99–101

Bibliographic databases:

UDC: 511

Citation: É. I. Kovalevskaya, “A geometric property of extremal surfaces”, Mat. Zametki, 23:2 (1978), 177–181; Math. Notes, 23:2 (1978), 99–101

Citation in format AMSBIB
\Bibitem{Kov78} \by \'E.~I.~Kovalevskaya \paper A~geometric property of extremal surfaces \jour Mat. Zametki \yr 1978 \vol 23 \issue 2 \pages 177--181 \mathnet{http://mi.mathnet.ru/mz8131} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=492661} \zmath{https://zbmath.org/?q=an:0406.10042|0387.10020} \transl \jour Math. Notes \yr 1978 \vol 23 \issue 2 \pages 99--101 \crossref{https://doi.org/10.1007/BF01153147} 

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This publication is cited in the following articles:
1. V. G. Sprindzhuk, “Achievements and problems in Diophantine approximation theory”, Russian Math. Surveys, 35:4 (1980), 1–80
2. E. I. Kovalevskaya, “Trigonometricheskie summy v metricheskoi teorii diofantovykh priblizhenii”, Chebyshevskii sb., 20:2 (2019), 207–220
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