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''in number theory''
 
''in number theory''
  
 
A method for obtaining non-trivial estimates of trigonometric sums (cf. [[Trigonometric sum|Trigonometric sum]]) of the form
 
A method for obtaining non-trivial estimates of trigonometric sums (cf. [[Trigonometric sum|Trigonometric sum]]) of the form
  
<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/w/w097/w097720/w0977201.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
S( f  )  = \sum _ {1 \leq  x \leq  P } e ^ {2 \pi i f( x) } ,
 +
$$
  
 
where
 
where
  
<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/w/w097/w097720/w0977202.png" /></td> </tr></table>
+
$$
 +
f( x)  = \alpha _ {n} x  ^ {n} + \dots + \alpha _ {1} x
 +
$$
  
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097720/w0977203.png" /> are arbitrary real numbers. Developed by H. Weyl [[#References|[1]]] to establish criteria for [[Uniform distribution|uniform distribution]] (cf. [[Weyl criterion|Weyl criterion]]).
+
and $  \alpha _ {n} \dots \alpha _ {1} $
 +
are arbitrary real numbers. Developed by H. Weyl [[#References|[1]]] to establish criteria for [[Uniform distribution|uniform distribution]] (cf. [[Weyl criterion|Weyl criterion]]).
  
The method may be described as follows. The sum (1) is raised to the power <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097720/w0977204.png" /> by successive squaring operations in order to reduce the degree of the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097720/w0977205.png" />. Thus, in the first stage
+
The method may be described as follows. The sum (1) is raised to the power $  \rho _ {0} = 2  ^ {n-} 1 $
 +
by successive squaring operations in order to reduce the degree of the polynomial $  f( x) $.  
 +
Thus, in the first stage
  
<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/w/w097/w097720/w0977206.png" /></td> </tr></table>
+
$$
 +
S  ^ {2} ( f  )  = \sum _ {\lambda _ {1} \neq 0 } \sum _ { x }
 +
e ^ {2 \pi i ( f( x+ \lambda _ {1} )- f( x)) } + O( P),
 +
$$
  
where the summations are performed over intervals of lengths <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097720/w0977207.png" />; now
+
where the summations are performed over intervals of lengths $  \ll  P $;  
 +
now
  
<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/w/w097/w097720/w0977208.png" /></td> </tr></table>
+
$$
 +
f( x+ \lambda _ {1} ) - f( x)  = \
 +
n \alpha _ {n} \lambda _ {1} x  ^ {n-} 1 + \dots
 +
$$
  
is a polynomial of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097720/w0977209.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097720/w09772010.png" /> (the symbols <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097720/w09772011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097720/w09772012.png" /> denote magnitudes of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097720/w09772013.png" />). At the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097720/w09772014.png" />-st step one obtains inner sum
+
is a polynomial of degree $  n - 1 $
 +
in $  x $(
 +
the symbols $  O( P) $,  
 +
$  \ll  P $
 +
denote magnitudes of order $  P $).  
 +
At the $  ( n - 1) $-
 +
st step one obtains inner sum
  
<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/w/w097/w097720/w09772015.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
S( \alpha )= \sum _ {x= a+ 1 } ^ { {a+ }  P _ {1} }
 +
e ^ {2 \pi i \alpha x } ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097720/w09772016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097720/w09772017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097720/w09772018.png" />. Sums of the form (2) are estimated using the inequality
+
where $  P _ {1} \leq  P $,  
 +
$  \alpha = n! \lambda _ {1} {} \dots \lambda _ {n-} 1 \alpha _ {n} $,  
 +
$  \lambda _ {i} \neq 0 $.  
 +
Sums of the form (2) are estimated using the inequality
  
<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/w/w097/w097720/w09772019.png" /></td> </tr></table>
+
$$
 +
| S( \alpha ) |  \leq  \min
 +
\left ( P _ {1} ,
 +
\frac{1}{| \sin  \pi \alpha | }
 +
\right ) ,
 +
$$
  
 
and the resulting estimate is:
 
and the resulting estimate is:
  
<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/w/w097/w097720/w09772020.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{3 }
 +
| S( f  ) | ^ {\rho _ {0} }  \leq  P ^ {\rho _ {0} - 1 } +
 +
$$
  
<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/w/w097/w097720/w09772021.png" /></td> </tr></table>
+
$$
 +
+
 +
P ^ {\rho _ {0} {- n + \epsilon } } \sum _ {0 < y \leq  P  ^ {n-} 1 } \min \left
 +
( P _ {1} ,
 +
\frac{1}{| \sin  \pi y \alpha _ {n} | }
 +
\right ) .
 +
$$
  
The inequality (3) yields different estimates for the sum (1) in case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097720/w09772022.png" /> is small in comparison to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097720/w09772023.png" />. These estimates depend on the accuracy with which the coefficient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097720/w09772024.png" /> of the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097720/w09772025.png" /> can be approximated by rational fractions.
+
The inequality (3) yields different estimates for the sum (1) in case $  1/ | \sin  \pi y \alpha _ {n} | $
 +
is small in comparison to $  P $.  
 +
These estimates depend on the accuracy with which the coefficient $  \alpha _ {n} $
 +
of the polynomial $  f( x) $
 +
can be approximated by rational fractions.
  
 
Example. Let
 
Example. Let
  
<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/w/w097/w097720/w09772026.png" /></td> </tr></table>
+
$$
 +
\left | \alpha _ {n} -  
 +
\frac{p}{q}
 +
\right |  \leq 
 +
\frac{1}{q  ^ {2} }
 +
,
 +
\  ( a, q) = 1.
 +
$$
  
 
Then
 
Then
  
<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/w/w097/w097720/w09772027.png" /></td> </tr></table>
+
$$
 +
| S( f  ) |  \ll  P ^ {1 + \epsilon } q  ^  \epsilon
 +
\left (
 +
\frac{1}{P}
 +
+
 +
\frac{1}{q}
 +
+
 +
\frac{q}{P  ^ {n} }
 +
\right ) ^ {\rho _ {0} } .
 +
$$
  
 
In particular, if
 
In particular, if
  
<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/w/w097/w097720/w09772028.png" /></td> </tr></table>
+
$$
 +
P  \leq  q  \leq  P  ^ {n-} 1 ,
 +
$$
  
 
then
 
then
  
<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/w/w097/w097720/w09772029.png" /></td> </tr></table>
+
$$
 +
| S( f) |  \ll  P ^ {1- \rho _ {0} + \epsilon } .
 +
$$
  
Weyl's method gives solutions, to a first approximation, of several important problems in number theory. The estimate (3) and its corollaries were used to study the [[Distribution modulo one|distribution modulo one]] of the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097720/w09772030.png" />. G.H. Hardy and J.E. Littlewood (1919) gave a solution of the [[Waring problem|Waring problem]] which was based on estimating the sums (1) by Weyl's method. They could thus estimate the values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097720/w09772031.png" /> for which the equation
+
Weyl's method gives solutions, to a first approximation, of several important problems in number theory. The estimate (3) and its corollaries were used to study the [[Distribution modulo one|distribution modulo one]] of the polynomial $  f( x) $.  
 +
G.H. Hardy and J.E. Littlewood (1919) gave a solution of the [[Waring problem|Waring problem]] which was based on estimating the sums (1) by Weyl's method. They could thus estimate the values of $  r= r( k) $
 +
for which the equation
  
<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/w/w097/w097720/w09772032.png" /></td> </tr></table>
+
$$
 +
N = x _ {1}  ^ {k} + \dots + x _ {r}  ^ {k} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097720/w09772033.png" /> is an integer and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097720/w09772034.png" /> are integers, is solvable, and even gave an asymptotic formula for the number of solutions. A generalization of the estimate (3) to the case of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097720/w09772035.png" /> which are not polynomials but are in a certain sense close to polynomials, resulted in the improvement of certain theorems in the theory of the [[Distribution of prime numbers|distribution of prime numbers]] (an estimate of the difference between two successive prime numbers and an estimate of the residual term in the asymptotic formula for the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097720/w09772036.png" /> of prime numbers not exceeding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097720/w09772037.png" />).
+
where $  N > 0 $
 +
is an integer and $  x _ {i} $
 +
are integers, is solvable, and even gave an asymptotic formula for the number of solutions. A generalization of the estimate (3) to the case of functions $  f( x) $
 +
which are not polynomials but are in a certain sense close to polynomials, resulted in the improvement of certain theorems in the theory of the [[Distribution of prime numbers|distribution of prime numbers]] (an estimate of the difference between two successive prime numbers and an estimate of the residual term in the asymptotic formula for the number $  \pi ( N) $
 +
of prime numbers not exceeding $  N $).
  
The insufficient strength of the estimates obtained by Weyl's method is due to the high power <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097720/w09772038.png" /> to which the sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097720/w09772039.png" /> is raised. J. van der Corput proposed a somewhat improved method for estimating the sums (1). The [[Vinogradov method|Vinogradov method]] yields a very accurate upper bound for the integral
+
The insufficient strength of the estimates obtained by Weyl's method is due to the high power $  \rho _ {0} $
 +
to which the sum $  S( f  ) $
 +
is raised. J. van der Corput proposed a somewhat improved method for estimating the sums (1). The [[Vinogradov method|Vinogradov method]] yields a very accurate upper bound for the 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/w/w097/w097720/w09772040.png" /></td> </tr></table>
+
$$
 +
\int\limits _ { 0 } ^ { 1 }  \dots \int\limits _ { 0 } ^ { 1 }
 +
| S( f  ) |  ^ {2b}  d \alpha _ {n} \dots d \alpha _ {1}  $$
  
already for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097720/w09772041.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097720/w09772042.png" /> a constant, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097720/w09772043.png" />). This estimate (cf. [[Vinogradov theorem about the average|Vinogradov theorem about the average]]) may be used to deduce essentially new estimates of Weyl sums (1) (with reduction factor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097720/w09772044.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097720/w09772045.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097720/w09772046.png" /> a constant), which cannot be attained by Weyl's method.
+
already for $  b \geq  cn  ^ {2} $(
 +
$  c > 0 $
 +
a constant, $  n \geq  2 $).  
 +
This estimate (cf. [[Vinogradov theorem about the average|Vinogradov theorem about the average]]) may be used to deduce essentially new estimates of Weyl sums (1) (with reduction factor $  P ^ {- \rho } $,  
 +
$  \rho = c _ {1} n  ^ {2}  \mathop{\rm ln}  n $;
 +
$  c _ {1} > 0 $
 +
a constant), which cannot be attained by Weyl's method.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Weyl,  "Ueber die Gleichverteilung von Zahlen mod. Eins"  ''Math. Ann.'' , '''77'''  (1916)  pp. 313–352</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.M. Vinogradov,  "The method of trigonometric sums in the theory of numbers" , Interscience  (1954)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Weyl,  "Ueber die Gleichverteilung von Zahlen mod. Eins"  ''Math. Ann.'' , '''77'''  (1916)  pp. 313–352</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.M. Vinogradov,  "The method of trigonometric sums in the theory of numbers" , Interscience  (1954)  (Translated from Russian)</TD></TR></table>

Latest revision as of 08:29, 6 June 2020


in number theory

A method for obtaining non-trivial estimates of trigonometric sums (cf. Trigonometric sum) of the form

$$ \tag{1 } S( f ) = \sum _ {1 \leq x \leq P } e ^ {2 \pi i f( x) } , $$

where

$$ f( x) = \alpha _ {n} x ^ {n} + \dots + \alpha _ {1} x $$

and $ \alpha _ {n} \dots \alpha _ {1} $ are arbitrary real numbers. Developed by H. Weyl [1] to establish criteria for uniform distribution (cf. Weyl criterion).

The method may be described as follows. The sum (1) is raised to the power $ \rho _ {0} = 2 ^ {n-} 1 $ by successive squaring operations in order to reduce the degree of the polynomial $ f( x) $. Thus, in the first stage

$$ S ^ {2} ( f ) = \sum _ {\lambda _ {1} \neq 0 } \sum _ { x } e ^ {2 \pi i ( f( x+ \lambda _ {1} )- f( x)) } + O( P), $$

where the summations are performed over intervals of lengths $ \ll P $; now

$$ f( x+ \lambda _ {1} ) - f( x) = \ n \alpha _ {n} \lambda _ {1} x ^ {n-} 1 + \dots $$

is a polynomial of degree $ n - 1 $ in $ x $( the symbols $ O( P) $, $ \ll P $ denote magnitudes of order $ P $). At the $ ( n - 1) $- st step one obtains inner sum

$$ \tag{2 } S( \alpha )= \sum _ {x= a+ 1 } ^ { {a+ } P _ {1} } e ^ {2 \pi i \alpha x } , $$

where $ P _ {1} \leq P $, $ \alpha = n! \lambda _ {1} {} \dots \lambda _ {n-} 1 \alpha _ {n} $, $ \lambda _ {i} \neq 0 $. Sums of the form (2) are estimated using the inequality

$$ | S( \alpha ) | \leq \min \left ( P _ {1} , \frac{1}{| \sin \pi \alpha | } \right ) , $$

and the resulting estimate is:

$$ \tag{3 } | S( f ) | ^ {\rho _ {0} } \leq P ^ {\rho _ {0} - 1 } + $$

$$ + P ^ {\rho _ {0} {- n + \epsilon } } \sum _ {0 < y \leq P ^ {n-} 1 } \min \left ( P _ {1} , \frac{1}{| \sin \pi y \alpha _ {n} | } \right ) . $$

The inequality (3) yields different estimates for the sum (1) in case $ 1/ | \sin \pi y \alpha _ {n} | $ is small in comparison to $ P $. These estimates depend on the accuracy with which the coefficient $ \alpha _ {n} $ of the polynomial $ f( x) $ can be approximated by rational fractions.

Example. Let

$$ \left | \alpha _ {n} - \frac{p}{q} \right | \leq \frac{1}{q ^ {2} } , \ ( a, q) = 1. $$

Then

$$ | S( f ) | \ll P ^ {1 + \epsilon } q ^ \epsilon \left ( \frac{1}{P} + \frac{1}{q} + \frac{q}{P ^ {n} } \right ) ^ {\rho _ {0} } . $$

In particular, if

$$ P \leq q \leq P ^ {n-} 1 , $$

then

$$ | S( f) | \ll P ^ {1- \rho _ {0} + \epsilon } . $$

Weyl's method gives solutions, to a first approximation, of several important problems in number theory. The estimate (3) and its corollaries were used to study the distribution modulo one of the polynomial $ f( x) $. G.H. Hardy and J.E. Littlewood (1919) gave a solution of the Waring problem which was based on estimating the sums (1) by Weyl's method. They could thus estimate the values of $ r= r( k) $ for which the equation

$$ N = x _ {1} ^ {k} + \dots + x _ {r} ^ {k} , $$

where $ N > 0 $ is an integer and $ x _ {i} $ are integers, is solvable, and even gave an asymptotic formula for the number of solutions. A generalization of the estimate (3) to the case of functions $ f( x) $ which are not polynomials but are in a certain sense close to polynomials, resulted in the improvement of certain theorems in the theory of the distribution of prime numbers (an estimate of the difference between two successive prime numbers and an estimate of the residual term in the asymptotic formula for the number $ \pi ( N) $ of prime numbers not exceeding $ N $).

The insufficient strength of the estimates obtained by Weyl's method is due to the high power $ \rho _ {0} $ to which the sum $ S( f ) $ is raised. J. van der Corput proposed a somewhat improved method for estimating the sums (1). The Vinogradov method yields a very accurate upper bound for the integral

$$ \int\limits _ { 0 } ^ { 1 } \dots \int\limits _ { 0 } ^ { 1 } | S( f ) | ^ {2b} d \alpha _ {n} \dots d \alpha _ {1} $$

already for $ b \geq cn ^ {2} $( $ c > 0 $ a constant, $ n \geq 2 $). This estimate (cf. Vinogradov theorem about the average) may be used to deduce essentially new estimates of Weyl sums (1) (with reduction factor $ P ^ {- \rho } $, $ \rho = c _ {1} n ^ {2} \mathop{\rm ln} n $; $ c _ {1} > 0 $ a constant), which cannot be attained by Weyl's method.

References

[1] H. Weyl, "Ueber die Gleichverteilung von Zahlen mod. Eins" Math. Ann. , 77 (1916) pp. 313–352
[2] I.M. Vinogradov, "The method of trigonometric sums in the theory of numbers" , Interscience (1954) (Translated from Russian)
How to Cite This Entry:
Weyl method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Weyl_method&oldid=49204
This article was adapted from an original article by B.M. Bredikhin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article