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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
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[1] [BM]Balasubramanian, R. &Mozzochi, C. J., An improved upper bound forG(k) in Waring’s problem for relatively smallk.Acta Arith., 43 (1984), 283–285. · Zbl 0491.10036
[2] [B1]de Bruijn, N. G., The asymptotic behavivour of a function occurring in the theory of primes.J. Indian Math. Soc. (N.S.), 15 (1951), 25–32.
[3] [B2], On the number of positive integers and free of prime factors >y Nederl. Acad. Wetensch. Proc. Ser. A., 54 (1951), 50–60. · Zbl 0042.04204
[4] [C]Chen, Jing-Run, On Waring’s problem fornth powers.Acta Math. Sinica, 8 (1958), 253–257; translated inChin. Math. Acta, 8 (1966), 849–853. · Zbl 0149.28904
[5] [D1]Davenport, H., On Waring’s problem for fourth powers.Ann. of Math., 40 (1939), 731–747. · JFM 65.1149.02
[6] [D2], On sums of positive integralkth powers.Amer. J. Math., 64 (1942), 189–198. · Zbl 0060.11908
[7] [D3]Davenport, H.,The collected works of Harold Davenport, vol. III. Ed. B. J. Birch, H., Halberstam & C. A. Rogers, Academic Press, 1977.
[8] [D4]Davenport, H. Multiplicative number theory. Springer-Verlag, second edition, 1980.
[9] [DE]Davenport, H. &Erdos, P., On sums of positive integralkth powers.Ann. of Math., 40 (1939), 533–536. · Zbl 0021.20703
[10] [DL]Davenport, H. &Lewis, D. J., Homogeneous additive equations.Proc. Roy. Soc. London Ser. A. 274 (1963), 443–460. · Zbl 0118.28002
[11] [E]Estermann, T., Einige Sätze über quadratfreie Zahlen.Math. Ann., 105 (1931), 653–662. · JFM 57.0222.02
[12] [HR]Halberstam, H. & Richert, H.-E.,Sieve methods. Academic Press, 1974. · Zbl 0298.10026
[13] [HL]Hardy, G. H. & Littlewood, J. E.,Collected papers of G. H. Hardy, including joint papers with J. E. Littlewood and others. Ed. by a committee appointed by the London Mathematical Society, vol. I, Clarendon Press, 1966.
[14] [H]Hooley, C., On Waring’s problem.Acta Math., 57 (1986), 49–97. · Zbl 0614.10038
[15] [K]Karatsuba, A. A., On the functionG(n) in Waring’s problem.Izv. Akad. Nauk SSSR, 49 1985), 935–947, 1119. · Zbl 0594.10041
[16] [L]Linnik, Ju. V., On the representation of large numbers as sums of seven cubes.Dokl. Akad. Nauk SSSR, 35 (1942), 162 andMat. Sb., 12 (1943), 218–224. · Zbl 0063.03579
[17] [T1]Thanigasalam, K., On Waring’s problem.Acta Arith., 38 (1980), 141–155. · Zbl 0368.10037
[18] [T2], Some new estimates forG(k) in Waring’s problem.Acta Arith. 42 (1982), 73–78. · Zbl 0496.10030
[19] [T3], Improvement on Davenport’s iterative method and new results in additive number theory, I & II, proof thatG(5)2.Acta Arith., 46 (1985), 1–31 and 91–112.
[20] [Va1]Vaughan, R. C., Homogeneous additive equations and Waring’s problem.Acta Arith., 33 (1977), 231–253. · Zbl 0361.10046
[21] [Va2]Vaughan, R. C.,The Hardy-Littlewood method. Cambridge University Press, 1981. · Zbl 0455.10034
[22] [Va3], Sums of three positive cubes,Bull. London Math. Soc, 17 (1985), 17–20. · Zbl 0562.10022
[23] [Va4], On Waring’s problem for smaller exponents.Proc. London Math. Soc. (3), 52 (1986), 445–63. · Zbl 0601.10035
[24] [Va5], On Waring’s problem for sixth powers.J. London Math. Soc. (2), 33 (1986), 227–236. · Zbl 0601.10036
[25] [Va6], On Waring’s problem for cubes.J. Reine Angew. Math. 365 (1986), 122–170. · Zbl 0574.10046
[26] [Va7], On Waring’s problem for smaller exponents II.Mathematika, 33 (1986), 6–22. · Zbl 0601.10037
[27] [Vi1]Vinogradov, I. M., The method of trigonometrical sums in the theory of numbers.Trudy Mat. Inst. Steklov, 23 (1947), 1–109.
[28] [Vi2], On an upper bound forG(n).Izv. Akad. Nauk SSSR, 23 (1959), 637–642. · Zbl 0089.02703
[29] [Vi3]Vinogradov, I. M.,Selected works. Springer-Verlag.
[30] [W]Watson, G. L., A proof of the seven cube theorem.J. London Math. Soc. 26 (1951), 153–156. · Zbl 0042.04101
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