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One of the most general methods in additive number theory. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c0222701.png" /> be arbitrary sets of natural numbers, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c0222702.png" /> be a natural number and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c0222703.png" /> be the number of solutions of the equation
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{{TEX|done}}{{MSC|11P55}}
  
<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/c/c022/c022270/c0222704.png" /></td> </tr></table>
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One of the most general methods in additive number theory. Let $X_1,\ldots,X_k$ be arbitrary sets of natural numbers, let $N$ be a natural number and let $J_k(N)$ be the number of solutions of the equation
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c0222705.png" />. It is with the investigation of the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c0222706.png" /> that additive number theory is concerned; for example, if it can be proved that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c0222707.png" /> is greater than zero for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c0222708.png" />, this means that any natural number is the sum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c0222709.png" /> terms taken respectively from the sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c02227010.png" />. Now let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c02227011.png" /> be a complex number, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c02227012.png" />, and let
+
$$ n_1+\cdots+n_k=N,$$
  
<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/c/c022/c022270/c02227013.png" /></td> </tr></table>
+
where $n_1\in X_1,\ldots,n_k\in X_k$. It is with the investigation of the numbers $J_k(N)$ that additive number theory is concerned; for example, if it can be proved that $J_k(N)$ is greater than zero for all $N$, this means that any natural number is the sum of $k$ terms taken respectively from the sets $X_1,\ldots,X_k$. Now let $s$ be a complex number and
  
Then the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c02227014.png" /> defined by
+
$$ g_m(s)=\sum_{n\in X_m}s^n.$$
  
<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/c/c022/c022270/c02227015.png" /></td> </tr></table>
+
Then the function $g(s)$ defined by
  
is the generating function of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c02227016.png" />. By Cauchy's formula,
+
$$ g(s)=g_1(s)\cdots g_k(s)=\sum_{N=1}^\infty J_k(N)s^N$$
  
<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/c/c022/c022270/c02227017.png" /></td> </tr></table>
+
is the generating function of the $J_n(N)$. By Cauchy's formula,
  
The integral in this equality is investigated as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c02227018.png" />. The circle of integration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c02227019.png" /> is divided into "major" and "minor" arcs, the centres of which are rational numbers. There is a broad range of additive problems in which the integrals over "major" arcs, which yield a "principal" part of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c02227020.png" />, can be investigated fairly completely, while the integrals over the "minor" arcs, which yield a "remainder" term in the asymptotic formula for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c02227021.png" />, can be estimated.
+
$$ J_k(N)=\frac{1}{2\pi i}\int_{\lvert s\rvert=R<1} g(s)s^{-(N+1)}\,\mathrm{d}s.$$
 +
 
 +
The integral in this equality is investigated as $R\to 1-0$. The circle of integration $\lvert s\rvert=R$ is divided into "major" and "minor" arcs, the centres of which are rational numbers. There is a broad range of additive problems in which the integrals over "major" arcs, which yield a "principal" part of $J_k(N)$, can be investigated fairly completely, while the integrals over the "minor" arcs, which yield a "remainder" term in the asymptotic formula for $J_k(N)$, can be estimated.
  
 
I.M. Vinogradov's use of trigonometric sums in the circle method not only considerably simplified application of the method, it also provided a unified approach to the solution of a wide range of very different additive problems. The basis for the circle method in the form of trigonometric sums is the formula
 
I.M. Vinogradov's use of trigonometric sums in the circle method not only considerably simplified application of the method, it also provided a unified approach to the solution of a wide range of very different additive problems. The basis for the circle method in the form of trigonometric sums is the formula
  
<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/c/c022/c022270/c02227022.png" /></td> </tr></table>
+
$$\int_0^1 e^{2\pi i\alpha m}\,\mathrm{d}\alpha=\begin{cases}1&\text{if }m=0,\\0&\text{if }m\neq0\text{ and $m$ an integer.}\end{cases}$$
  
 
It follows from this formula that
 
It follows from this formula that
  
<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/c/c022/c022270/c02227023.png" /></td> </tr></table>
+
$$ J_k(N)=\int_0^1 s_1(\alpha)\cdots s_k(\alpha)e^{-2\pi i\alpha N}\,\mathrm{d}\alpha,$$
  
 
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/c/c022/c022270/c02227024.png" /></td> </tr></table>
+
$$ s_m(\alpha)=\sum_{\substack{n\in X_m\\ n\leq N}}e^{2\pi i\alpha n},\quad m=1,\ldots,k.$$
  
The finite sums <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c02227025.png" /> are called trigonometric sums. To investigate the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c02227026.png" />, one divides the integration interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c02227027.png" /> into "major" and "minor" arcs, i.e. intervals centred at rational points with "small" and "large" denominators. For many additive problems one can successfully evaluate — with adequate accuracy — the integrals over the "major" arcs (the trigonometric sums for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c02227028.png" /> in "major" arcs are close to rational trigonometric sums with small denominators, which are readily evaluated and are "large" ); as for the "minor" arcs, which contain the bulk of the points in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c02227029.png" />, the trigonometric sums over these are "small" ; they can be estimated in a non-trivial manner (see [[Trigonometric sums, method of|Trigonometric sums, method of]]; [[Vinogradov method|Vinogradov method]]), so that asymptotic formulas can be established for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c02227030.png" />.
+
The finite sums $s_m(\alpha)$ are called trigonometric sums. To investigate the $J_k(N)$, one divides the integration interval $[0,1]$ into "major" and "minor" arcs, i.e. intervals centred at rational points with "small" and "large" denominators. For many additive problems one can successfully evaluate — with adequate accuracy — the integrals over the "major" arcs (the trigonometric sums for $\alpha$ in "major" arcs are close to rational trigonometric sums with small denominators, which are readily evaluated and are "large" ); as for the "minor" arcs, which contain the bulk of the points in $[0,1]$, the trigonometric sums over these are "small" ; they can be estimated in a non-trivial manner (see [[Trigonometric sums, method of|Trigonometric sums, method of]]; [[Vinogradov method|Vinogradov method]]), so that asymptotic formulas can be established for $J_k(N)$.
  
 
The circle method in the trigonometric sum version, together with Vinogradov's method for estimating trigonometric sums, yields the strongest results of additive number theory (see [[Waring problem|Waring problem]]; [[Goldbach problem|Goldbach problem]]; [[Goldbach–Waring problem|Goldbach–Waring problem]]; [[Hilbert–Kamke problem|Hilbert–Kamke problem]]).
 
The circle method in the trigonometric sum version, together with Vinogradov's method for estimating trigonometric sums, yields the strongest results of additive number theory (see [[Waring problem|Waring problem]]; [[Goldbach problem|Goldbach problem]]; [[Goldbach–Waring problem|Goldbach–Waring problem]]; [[Hilbert–Kamke problem|Hilbert–Kamke problem]]).
  
====References====
+
The circle method as described above is often referred to as the Hardy–Littlewood method or the Hardy–Littlewood circle method. The method can be adapted to a number of quite diverse situations. Some examples follow.
<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,  "The method of trigonometric sums and its applications to number theory" , ''Selected papers'' , Springer  (1983)  pp. 124–135  (Translated from German)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.A. Karatsuba,  "Fundamentals of analytic number theory" , Moscow  (1975)  (In Russian)</TD></TR></table>
 
  
 +
The Davenport–Heilbronn theorem says that if $\lambda_1,\ldots,\lambda_s$, $s\geq 2^k+1$, are real numbers, not all of the same sign if $k$ is even, and such that at least one ratio $\lambda_i/\lambda_j$ is irrational, then for all $\eta\geq0$ there are integers $x_1,\ldots,x_s$, not all zero, such that $\lvert x_1\lambda_1+\cdots+x_s\lambda_s\rvert\leq \eta$.
  
 +
Let $\mathcal{A}$ be a subset of the natural numbers such that $d(\mathcal{A})>0$, where $d(\mathcal{A})$ is the upper [[Asymptotic density|asymptotic density]]. Then the Furstenberg–Sárközy theorem says that if $R(n)$ is the number of solutions of $a-a'=x^2$ with $a,a'\in\mathcal{A}$, $a<n$, $x\in\N$, then $\lim_{n\to\infty}n^{-3/2}R(n)=0$.
  
====Comments====
+
Finally there is e.g. Birch's theorem to the effect that the dimension of the space of simultaneous zeros of $k$ homogeneous forms of odd degree grows arbitrarily large with the number of variables of those forms.
The circle method as described above is often referred to as the Hardy–Littlewood method or the Hardy–Littlewood circle method. The method can be adapted to a number of quite diverse situations. Some examples follow. The Davenport–Heilbron theorem says that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c02227031.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c02227032.png" />, are real numbers, not all of the same sign if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c02227033.png" /> is even, and such that at least one ratio <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c02227034.png" /> is irrational, then for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c02227035.png" /> there are integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c02227036.png" />, not all zero, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c02227037.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c02227038.png" /> be a subset of the natural numbers such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c02227039.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c02227040.png" /> is the upper [[Asymptotic density|asymptotic density]]. Then the Furstenberg–Sárközy theorem says that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c02227041.png" /> is the number of solutions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c02227042.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c02227043.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c02227044.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c02227045.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c02227046.png" />. Finally there is e.g. Birch's theorem to the effect that the dimension of the space of simultaneous zeros of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c02227047.png" /> homogeneous forms of odd degree grows arbitrarily large with the number of variables of those forms.
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"R.C. Vaughan,   "The Hardy–Littlewood method" , Cambridge Univ. Press (1981)</TD></TR></table>
+
{|
 +
|-
 +
|valign="top"|{{Ref|Hu}}||valign="top"| L.-K. Hua, "The method of trigonometric sums and its applications to number theory" , ''Selected papers'' , Springer (1983) pp. 124–135 (Translated from German)
 +
|-
 +
|valign="top"|{{Ref|Ka}}||valign="top"| A.A. Karatsuba, "Fundamentals of analytic number theory" , Moscow (1975) (In Russian)
 +
|-
 +
|valign="top"|{{Ref|Va}}||valign="top"| R.C. Vaughan, "The Hardy–Littlewood method" , Cambridge Univ. Press (1981)
 +
|-
 +
|valign="top"|{{Ref|Vi}}||valign="top"| I.M. Vinogradov, "The method of trigonometric sums in the theory of numbers" , Interscience (1954) (Translated from Russian)
 +
|-
 +
|}

Latest revision as of 21:24, 18 November 2016

2020 Mathematics Subject Classification: Primary: 11P55 [MSN][ZBL]

One of the most general methods in additive number theory. Let $X_1,\ldots,X_k$ be arbitrary sets of natural numbers, let $N$ be a natural number and let $J_k(N)$ be the number of solutions of the equation

$$ n_1+\cdots+n_k=N,$$

where $n_1\in X_1,\ldots,n_k\in X_k$. It is with the investigation of the numbers $J_k(N)$ that additive number theory is concerned; for example, if it can be proved that $J_k(N)$ is greater than zero for all $N$, this means that any natural number is the sum of $k$ terms taken respectively from the sets $X_1,\ldots,X_k$. Now let $s$ be a complex number and

$$ g_m(s)=\sum_{n\in X_m}s^n.$$

Then the function $g(s)$ defined by

$$ g(s)=g_1(s)\cdots g_k(s)=\sum_{N=1}^\infty J_k(N)s^N$$

is the generating function of the $J_n(N)$. By Cauchy's formula,

$$ J_k(N)=\frac{1}{2\pi i}\int_{\lvert s\rvert=R<1} g(s)s^{-(N+1)}\,\mathrm{d}s.$$

The integral in this equality is investigated as $R\to 1-0$. The circle of integration $\lvert s\rvert=R$ is divided into "major" and "minor" arcs, the centres of which are rational numbers. There is a broad range of additive problems in which the integrals over "major" arcs, which yield a "principal" part of $J_k(N)$, can be investigated fairly completely, while the integrals over the "minor" arcs, which yield a "remainder" term in the asymptotic formula for $J_k(N)$, can be estimated.

I.M. Vinogradov's use of trigonometric sums in the circle method not only considerably simplified application of the method, it also provided a unified approach to the solution of a wide range of very different additive problems. The basis for the circle method in the form of trigonometric sums is the formula

$$\int_0^1 e^{2\pi i\alpha m}\,\mathrm{d}\alpha=\begin{cases}1&\text{if }m=0,\\0&\text{if }m\neq0\text{ and $m$ an integer.}\end{cases}$$

It follows from this formula that

$$ J_k(N)=\int_0^1 s_1(\alpha)\cdots s_k(\alpha)e^{-2\pi i\alpha N}\,\mathrm{d}\alpha,$$

where

$$ s_m(\alpha)=\sum_{\substack{n\in X_m\\ n\leq N}}e^{2\pi i\alpha n},\quad m=1,\ldots,k.$$

The finite sums $s_m(\alpha)$ are called trigonometric sums. To investigate the $J_k(N)$, one divides the integration interval $[0,1]$ into "major" and "minor" arcs, i.e. intervals centred at rational points with "small" and "large" denominators. For many additive problems one can successfully evaluate — with adequate accuracy — the integrals over the "major" arcs (the trigonometric sums for $\alpha$ in "major" arcs are close to rational trigonometric sums with small denominators, which are readily evaluated and are "large" ); as for the "minor" arcs, which contain the bulk of the points in $[0,1]$, the trigonometric sums over these are "small" ; they can be estimated in a non-trivial manner (see Trigonometric sums, method of; Vinogradov method), so that asymptotic formulas can be established for $J_k(N)$.

The circle method in the trigonometric sum version, together with Vinogradov's method for estimating trigonometric sums, yields the strongest results of additive number theory (see Waring problem; Goldbach problem; Goldbach–Waring problem; Hilbert–Kamke problem).

The circle method as described above is often referred to as the Hardy–Littlewood method or the Hardy–Littlewood circle method. The method can be adapted to a number of quite diverse situations. Some examples follow.

The Davenport–Heilbronn theorem says that if $\lambda_1,\ldots,\lambda_s$, $s\geq 2^k+1$, are real numbers, not all of the same sign if $k$ is even, and such that at least one ratio $\lambda_i/\lambda_j$ is irrational, then for all $\eta\geq0$ there are integers $x_1,\ldots,x_s$, not all zero, such that $\lvert x_1\lambda_1+\cdots+x_s\lambda_s\rvert\leq \eta$.

Let $\mathcal{A}$ be a subset of the natural numbers such that $d(\mathcal{A})>0$, where $d(\mathcal{A})$ is the upper asymptotic density. Then the Furstenberg–Sárközy theorem says that if $R(n)$ is the number of solutions of $a-a'=x^2$ with $a,a'\in\mathcal{A}$, $a<n$, $x\in\N$, then $\lim_{n\to\infty}n^{-3/2}R(n)=0$.

Finally there is e.g. Birch's theorem to the effect that the dimension of the space of simultaneous zeros of $k$ homogeneous forms of odd degree grows arbitrarily large with the number of variables of those forms.

References

[Hu] L.-K. Hua, "The method of trigonometric sums and its applications to number theory" , Selected papers , Springer (1983) pp. 124–135 (Translated from German)
[Ka] A.A. Karatsuba, "Fundamentals of analytic number theory" , Moscow (1975) (In Russian)
[Va] R.C. Vaughan, "The Hardy–Littlewood method" , Cambridge Univ. Press (1981)
[Vi] I.M. Vinogradov, "The method of trigonometric sums in the theory of numbers" , Interscience (1954) (Translated from Russian)
How to Cite This Entry:
Circle method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Circle_method&oldid=16932
This article was adapted from an original article by A.A. Karatsuba (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article