Waring problem

2020 Mathematics Subject Classification: Primary: 11P05 [MSN][ZBL]

A problem in number theory formulated in 1770 by E. Waring in the following form: Any natural number is a sum of 4 squares, of 9 cubes and of 19 fourth-powers. In other words, for all $n\geq2$ there exists a $k=k(n)$, depending only on $n$, such that every natural number is the sum of $k$ $n$-th powers of non-negative integers. D. Hilbert in 1909 was the first to give a general solution of Waring's problem with a very rough estimate of the value of $k$ as a function of $n$; this is why the problem is sometimes known as the Hilbert–Waring problem. Let $J_{k,n}(N)$ be the number of solutions of the equation

\begin{equation}\label{war}x_1^n+\cdots+x_k^n=N\end{equation}

in non-negative integers. Hilbert's theorem then states that there exists a $K=k(n)$ for which $J_{K,n}(N)\geq1$ for any $N\geq1$. G.H. Hardy and J.E. Littlewood, who applied the circle method to the Waring problem, demonstrated in 1928 that for $k\geq(n-2)2^{n-1}+5$ the value of $J_{k,n}(N)$ is given by an asymptotic formula of the type

\begin{equation}\label{asym}J_{k,n}(N)=AN^{k/n-1}+O(N^{k/n-1-\gamma}),\end{equation}

where $A=A(N)\geq c_0>0$, while $c_0$ and $\gamma>0$ are constants. Consequently, if $N\geq N_0(n)$, equation \ref{war} has a solution.

An elementary proof of Waring's problem was given in 1942 by Yu.V. Linnik. There exist many different generalizations of Waring's problem (the variables run through a certain subset of the set of natural numbers; the number $N$ is represented by polynomials $f_1(x_1),\ldots,f_k(x_k)$ rather than by monomials $x_1^n,\ldots,x_k^n$; equation (1) is replaced by a congruence, etc.).

Research on Waring's problem has mainly focused on sharpening estimates for the following three questions:

1. Find the smallest $k$ such that \ref{war} has solutions for all sufficiently large $N$;
2. Find the smallest $k$ such that \ref{war} has solutions for all $N$;
3. Find the smallest $k$ such that the number of solutions to \ref{war}, $J_{k,n}(N)$, is given by the asymptotic formula \ref{asym}.

These quantities are known as $G(n)$, $g(n)$, and $k_0(n)$ respectively. Clearly, $k_0(n)\geq G(n)$ and $g(n)\geq G(n)$. The progress on bounds for these quantities is detailed below.

Solvable for $N$ sufficiently large

Let $G(n)$ be the smallest integer such that equation \ref{war} is solvable for $k\geq G(n)$ and $N$ sufficiently large depending on $n$.

It is known that $G(n)\geq n+1$. It was proved in 1934 by I.M. Vinogradov, using his own method, that

$$G(n)\leq 3n(\ln n+9).$$

Moreover, many results are available concerning $G(n)$ for small values of $n$: $G(4)=16$ (H. Davenport, 1939); $G(3)=7$ (Yu.V. Linnik, 1942).

Solvable for all $N$

Let $g(n)$ be the smallest integer such that equation \ref{war} is solvable for $k\geq g(n)$ and $N\geq1$.

It was shown in 1936 by L. Dickson and S. Pillai, who also used the Vinogradov method, that

$$g(n)=2^n+\left[\left(\frac{3}{2}\right)^n\right]-2$$

for all $n>6$ for which

$$\left(\frac{3}{2}\right)^n-\left[\left(\frac{3}{2}\right)^n\right]\leq1-\left(\frac{1}{2}\right)^n\left\{\left[\left(\frac{3}{2}\right)^n\right]+2\right\}.$$

The last condition was demonstrated in 1957 by K. Mahler for all sufficiently large $n$.

It is known that $g(2)=4$ (J.L. Lagrange, 1770), $g(3)=9$ (A. Wieferich, A. Kempner, 1912), $g(4)=19$ (R. Balusabramanian, J. Deshouillers, F. Dress, 1986), $g(5)=37$ (Chen-Jingrun, 1964). See also Circle method and [HaWr]–[Sh].

Asymptotic formula

Let $\tilde{G}(k)$ be the smallest integer such that the asymptotic formula \ref{asym} applies to $J_{s,k}(N)$ if $s\geq \tilde{G}(k)$. The result of Hardy and Littlewood mentioned above shows that

$$\tilde{G}(k)\leq(k-2)2^{k-1}+5.$$

The first substantial improvement for large values of $k$ was obtained by Vinogradov, who showed that

$$\tilde{G}(k)\leq 4k^2\ln k.$$

The current best bound for large values of $k$ was obtained by Wooley who showed that

$$\tilde{G}(k)\leq 2k^2-k^{4/3}+O(k).$$

References

[De]	B.N. Delone, "The Peterburg school of number theory", Moscow-Leningrad (1947) (In Russian)
[Hu]	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)
[Kh]	A.Ya. Khinchin, "Three pearls of number theory", Graylock (1952) (Translated from Russian)
[Vi]	I.M. Vinogradov, "Selected works", Springer (1985) (Translated from Russian)
[Vi2]	I.M. Vinogradov, "The method of trigonometric sums in the theory of numbers", Interscience (1954) (Translated from Russian)
[HaWr]	G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers", Oxford Univ. Press (1979) pp. Chapt. 6
[Sh]	D. Shanks, "Solved and unsolved problems in number theory", Chelsea, reprint (1978)
[Va]	R.C. Vaughan, "The Hardy–Littlewood method", Cambridge Univ. Press (1981)
[Wo]	T. D. Wooley, "Vinogradov's mean value theorem via efficient congruencing", Annals of Math. 175 (2012), 1575--1627.