## A new iterative method in Waring’s problem.(English)Zbl 0665.10033

This paper constitutes the most significant contribution to the theory of Waring’s theorem in recent years, and improves on almost all what has been known on the subject. Take G(k) to be the smallest s such that every large natural number is the sum of s k-th powers of natural numbers. The author shows that G(5)$$\leq 19$$, G(6)$$\leq 29$$, G(7)$$\leq 41$$, G(8)$$\leq 58$$, and for large k, $G(k)<2k(\log k+\log \log k+1+\log 2+O(\frac{\log \log k}{\log k})).$ There are also improvements on G(k) for intermediate values of k, too long to be stated here, and when $$k=4$$ it is shown that all numbers satisfying a necessary congruence condition are sums of 12 biquadrates. Moreover the paper gives lower bounds for the number $$N_{s,k}(X)$$ of numbers not exceeding X which are sums of s k-th powers. When $$s\geq 3$$, $$k\geq 3$$, these are always superior to previously known bounds. As an example of particular interest it is shown that $$N_{3,3}(X)\gg X^{11/12-\epsilon}$$ which should be compared with the author’s earlier result [J. Reine Angew. Math. 365, 122-170 (1986; Zbl 0574.10046)] where the exponent 19/21-$$\epsilon$$ occurs in the place of 11/12-$$\epsilon$$.
Let $${\mathcal A}(P,R)=\{n\leq P:$$ $$p| n\Rightarrow p\leq R\}$$ and $$S_ s(P,R)$$ be the number of solutions of $x^ k_ 1+...+x^ k_ s=y^ k_ 1+...+y^ k_ s$ subject to $$x_ i,y_ i\in {\mathcal A}(P,R)$$. The main object of the paper is giving upper bounds for $$S_ s(P,P^{\epsilon})$$ (essentially), and this is an entirely new idea. The author first relates $$S_ s(P,R)$$ with $$T_ s(P,R,\vartheta)$$ where $$T_ s(P,R,\vartheta)$$ is the number of solutions of $(1)\quad x^ k+m^ k(x^ k_ 1+...+x^ k_{s-1})=y^ k+m^ k(y^ k_ 1+...+y^ k_{s-1})$ subject to x,y$$\leq P$$, $$x\equiv y mod m^ k$$, $$P^{\vartheta}\leq m\leq P^{\vartheta}R$$, $$x_ i,y_ i\in {\mathcal A}(P^{1-\vartheta},R)$$ with $$0<\vartheta <1/k$$ and $$P^{\vartheta}R<P$$. The quantity $$T_ s(P,R,\vartheta)$$ is more familiar in Waring’s problem, but it should be observed that in (1) homogeneity in the variables $$x_ i$$, $$y_ i$$ is conserved. This is a fundamental advantage to earlier methods. Existing techniques due to Davenport [see chapter 6 of the author’s monograph, The Hardy-Littlewood method (Cambridge, 1981; Zbl 0455.10034) and the author, Proc. Lond. Math. Soc., III. Ser. 52, 445-463 (1986; Zbl 0601.10035)] may now be used to relate $$T_ s(P,R,\vartheta)$$ with $$S_{s-1}(P^{1-\vartheta},R),$$ but the conservation of homogeneity allows one also to relate $$T_ s(P,R,\vartheta)$$ with $$S_ t(P^{1-\vartheta},R)$$ where t now differs from $$s-1.$$ The author describes various ways of doing this, and $$t=s$$ turns out to be an effective choice. One is now in a powerful position. Since $$S_ s(P,R)$$ can be bounded in terms of $$S_ s(P^{1- \vartheta},R)$$ and $$S_{s-1}(P^{1-\vartheta},R),$$ say, one can easily determine a bound of the shape $$S_ s(P,R)\ll P^{\lambda_ s}$$ provided $$\lambda_{s-1}$$ has already been computed. This establishes an iterative method of an entirely new kind.
The new results on Waring’s problem are deduced from the bounds on $$S_ s(P,R)$$ via the circle method. However there are other applications as well. By means of the large sieve the exponential sum $f(\alpha)=\sum_{x\in A(P,P^{\epsilon})}e^{2\pi i\alpha x^ k}$ can be bounded in terms of $$S_ s(P,P^{\epsilon})$$. Vaughan’s result is too complicated to be stated here, but for large k it reads $f(\alpha)\ll P^{1+\epsilon -(\frac{1}{4k \log k})(1+o(1))}$ where $$\alpha$$ is in a certain set of “minor arcs”. Such estimates might well be useful outside additive number theory.
Reviewer: J.Brüdern

### MSC:

 11P05 Waring’s problem and variants 11P55 Applications of the Hardy-Littlewood method 11L40 Estimates on character sums 11N35 Sieves

### Citations:

Zbl 0574.10046; Zbl 0455.10034; Zbl 0601.10035
Full Text: