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Difference between revisions of "Vinogradov theorem about the average"

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A theorem providing an upper bound of the value of a [[Vinogradov integral|Vinogradov integral]]:
 
A theorem providing an upper bound of the value of a [[Vinogradov integral|Vinogradov integral]]:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096700/v0967001.png" /></td> </tr></table>
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$$ J_b = J_{b, n} (P) = \int \limits_0^1 \dots \int \limits_0^1 \left| \sum_{x \, = \, 1}^P e^{2 \pi i (\alpha_n x^n + \dots + \alpha_1 x)} \right|^{2b} d \alpha_n \dots d \alpha_1, $$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096700/v0967002.png" /></td> </tr></table>
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where $J_b$ is the average value of the trigonometric sum. It is formulated as follows. If, for a non-negative integer $t$ one sets
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096700/v0967003.png" /> is the average value of the trigonometric sum. It is formulated as follows. If, for a non-negative integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096700/v0967004.png" /> one sets
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$$ D_t = (20n)^{n(n + 1)t/2}, \qquad b_t = nt + \left[{\frac{n(n + 1)}{4} + 1}\right], $$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096700/v0967005.png" /></td> </tr></table>
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then, if $l > 0$ and for an integer $b \geq b_l$,
  
then, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096700/v0967006.png" /> and for an integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096700/v0967007.png" />,
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$$ J_b = J_{b, n} (P) < D_l P^{2b - (1 + (1 - 1/n)^l)n(n + 1)/2}. $$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096700/v0967008.png" /></td> </tr></table>
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The estimate of $J_b$ given by Vinogradov's theorem is asymptotically exact. The theorem is fundamental in the [[Vinogradov method|Vinogradov method]] for estimating Weyl sums (cf. [[Weyl sum|Weyl sum]]). In addition, it yielded a number of almost optimal results in classical problems in number theory (cf. [[Waring problem|Waring problem]]; [[Hilbert–Kamke problem|Hilbert–Kamke problem]]; [[Distribution modulo one|Distribution modulo one]] of a polynomial).
 
 
The estimate of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096700/v0967009.png" /> given by Vinogradov's theorem is asymptotically exact. The theorem is fundamental in the [[Vinogradov method|Vinogradov method]] for estimating Weyl sums (cf. [[Weyl sum|Weyl sum]]). In addition, it yielded a number of almost optimal results in classical problems in number theory (cf. [[Waring problem|Waring problem]]; [[Hilbert–Kamke problem|Hilbert–Kamke problem]]; [[Distribution modulo one|Distribution modulo one]] of a polynomial).
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.M. Vinogradov,  "The method of trigonometric sums in the theory of numbers" , Interscience  (1954)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.-K. Hua,  "Abschätzungen von Exponentialsummen und ihre Anwendung in der Zahlentheorie" , ''Enzyklopaedie der Mathematischen Wissenschaften mit Einschluss ihrer Anwendungen'' , '''1''' :  2  (1959)  (Heft 13, Teil 1)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.M. Vinogradov,  "The method of trigonometric sums in the theory of numbers" , Interscience  (1954)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.-K. Hua,  "Abschätzungen von Exponentialsummen und ihre Anwendung in der Zahlentheorie" , ''Enzyklopaedie der Mathematischen Wissenschaften mit Einschluss ihrer Anwendungen'' , '''1''' :  2  (1959)  (Heft 13, Teil 1)</TD></TR></table>
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Latest revision as of 21:55, 14 January 2017

A theorem providing an upper bound of the value of a Vinogradov integral:

$$ J_b = J_{b, n} (P) = \int \limits_0^1 \dots \int \limits_0^1 \left| \sum_{x \, = \, 1}^P e^{2 \pi i (\alpha_n x^n + \dots + \alpha_1 x)} \right|^{2b} d \alpha_n \dots d \alpha_1, $$

where $J_b$ is the average value of the trigonometric sum. It is formulated as follows. If, for a non-negative integer $t$ one sets

$$ D_t = (20n)^{n(n + 1)t/2}, \qquad b_t = nt + \left[{\frac{n(n + 1)}{4} + 1}\right], $$

then, if $l > 0$ and for an integer $b \geq b_l$,

$$ J_b = J_{b, n} (P) < D_l P^{2b - (1 + (1 - 1/n)^l)n(n + 1)/2}. $$

The estimate of $J_b$ given by Vinogradov's theorem is asymptotically exact. The theorem is fundamental in the Vinogradov method for estimating Weyl sums (cf. Weyl sum). In addition, it yielded a number of almost optimal results in classical problems in number theory (cf. Waring problem; Hilbert–Kamke problem; Distribution modulo one of a polynomial).

References

[1] I.M. Vinogradov, "The method of trigonometric sums in the theory of numbers" , Interscience (1954) (Translated from Russian)
[2] L.-K. Hua, "Abschätzungen von Exponentialsummen und ihre Anwendung in der Zahlentheorie" , Enzyklopaedie der Mathematischen Wissenschaften mit Einschluss ihrer Anwendungen , 1 : 2 (1959) (Heft 13, Teil 1)
How to Cite This Entry:
Vinogradov theorem about the average. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Vinogradov_theorem_about_the_average&oldid=12231
This article was adapted from an original article by A.A. Karatsuba (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article