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Difference between revisions of "Circle method"

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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
 
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
  
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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 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 $\lambda_1,\ldots,\lambda_s$, $s\geq 2^k+1$, 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 $\lvert x_1\lambda_1+\cdots+x_s\lambda_s\rvert\leq \eta$.
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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 $\lim_{n\to\infty}n^{-3/2}R(n)=0$.
 
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 $\lim_{n\to\infty}n^{-3/2}R(n)=0$.

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=39763
This article was adapted from an original article by A.A. Karatsuba (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article