Circle method

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–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 $\lvert x_1\lambda_1+\cdots+x_s\lambda_s\rvert\leq \eta$.

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)