Home On Waring’s problem for larger powers
Article
Licensed
Unlicensed Requires Authentication

On Waring’s problem for larger powers

  • Jörg Brüdern EMAIL logo and Trevor D. Wooley ORCID logo
Published/Copyright: November 1, 2023

Abstract

Let G ( k ) denote the least number s having the property that every sufficiently large natural number is the sum of at most s positive integral k-th powers. Then for all k , one has

G ( k ) k ( log k + 4.20032 ) .

Our new methods improve on all bounds available hitherto when k 14 .

Award Identifier / Grant number: DMS-1854398

Award Identifier / Grant number: DMS-2001549

Award Identifier / Grant number: 255083470

Funding statement: The first author was supported by Deutsche Forschungsgemeinschaft Project Number 255083470. The second author was supported by NSF grants DMS-1854398 and DMS-2001549.

References

[1] J. Brüdern, A problem in additive number theory, Math. Proc. Cambridge Philos. Soc. 103 (1988), no. 1, 27–33. 10.1017/S0305004100064586Search in Google Scholar

[2] J. Brüdern and T. D. Wooley, Partitio numerorum: Sums of a prime and a number of k-th powers, preprint (2022), https://arxiv.org/abs/2211.10387. Search in Google Scholar

[3] G. H. Hardy and J. E. Littlewood, A new solution of Waring’s problem, Quart. J. Math. Oxford 48 (1920), 272–293. Search in Google Scholar

[4] G. H. Hardy and J. E. Littlewood, Some problems of “Partitio Numerorum”: IV. The singular series in Waring’s Problem and the value of the number G ( k ) , Math. Z. 12 (1922), no. 1, 161–188. 10.1007/BF01482074Search in Google Scholar

[5] G. H. Hardy and J. E. Littlewood, Some problems of “Partitio numerorum” (VI): Further researches in Waring’s Problem, Math. Z. 23 (1925), no. 1, 1–37. 10.1007/BF01506218Search in Google Scholar

[6] D. R. Heath-Brown, The fractional part of α n k , Mathematika 35 (1988), no. 1, 28–37. 10.1112/S0025579300006240Search in Google Scholar

[7] A. A. Karatsuba, The function G ( n ) in Waring’s problem, Izv. Akad. Nauk SSSR Ser. Mat. 49 (1985), no. 5, 935–947, 1119. Search in Google Scholar

[8] A. A. Karatsuba, On a Diophantine inequality, Acta Arith. 53 (1989), no. 3, 309–324. 10.4064/aa-53-3-309-324Search in Google Scholar

[9] J. Liu and L. Zhao, Representation by sums of unlike powers, J. reine angew. Math. 781 (2021), 19–55. 10.1515/crelle-2021-0048Search in Google Scholar

[10] S. T. Parsell and T. D. Wooley, Exceptional sets for Diophantine inequalities, Int. Math. Res. Not. IMRN 2014 (2014), no. 14, 3919–3974. 10.1093/imrn/rnt062Search in Google Scholar

[11] R. C. Vaughan, On Waring’s problem for smaller exponents, Proc. Lond. Math. Soc. (3) 52 (1986), no. 3, 445–463. 10.1112/plms/s3-52.3.445Search in Google Scholar

[12] R. C. Vaughan, A new iterative method in Waring’s problem, Acta Math. 162 (1989), no. 1–2, 1–71. 10.1007/BF02392834Search in Google Scholar

[13] R. C. Vaughan, The Hardy–Littlewood method, 2nd ed., Cambridge Tracts in Math. 125, Cambridge University, Cambridge 1997. 10.1017/CBO9780511470929Search in Google Scholar

[14] R. C. Vaughan and T. D. Wooley, On Waring’s problem: Some refinements, Proc. Lond. Math. Soc. (3) 63 (1991), no. 1, 35–68. 10.1112/plms/s3-63.1.35Search in Google Scholar

[15] R. C. Vaughan and T. D. Wooley, Further improvements in Waring’s problem, II: Sixth powers, Duke Math. J. 76 (1994), no. 3, 683–710. 10.1215/S0012-7094-94-07626-6Search in Google Scholar

[16] R. C. Vaughan and T. D. Wooley, Further improvements in Waring’s problem, Acta Math. 174 (1995), no. 2, 147–240. 10.1007/BF02392467Search in Google Scholar

[17] R. C. Vaughan and T. D. Wooley, Further improvements in Waring’s problem, IV: Higher powers, Acta Arith. 94 (2000), no. 3, 203–285. 10.4064/aa-94-3-203-285Search in Google Scholar

[18] I. M. Vinogradov, On an upper bound for G ( n ) , Izv. Akad. Nauk SSSR Ser. Mat. 23 (1959), 637–642. Search in Google Scholar

[19] T. D. Wooley, On simultaneous additive equations and Waring’s problem, Ph.D. thesis, University of London, 1990. 10.1112/S0025579300012821Search in Google Scholar

[20] T. D. Wooley, Large improvements in Waring’s problem, Ann. of Math. (2) 135 (1992), no. 1, 131–164. 10.2307/2946566Search in Google Scholar

[21] T. D. Wooley, The application of a new mean value theorem to the fractional parts of polynomials, Acta Arith. 65 (1993), no. 2, 163–179. 10.4064/aa-65-2-163-179Search in Google Scholar

[22] T. D. Wooley, New estimates for smooth Weyl sums, J. Lond. Math. Soc. (2) 51 (1995), no. 1, 1–13. 10.1112/jlms/51.1.1Search in Google Scholar

[23] T. D. Wooley, Rational solutions of pairs of diagonal equations, one cubic and one quadratic, Proc. Lond. Math. Soc. (3) 110 (2015), no. 2, 325–356. 10.1112/plms/pdu054Search in Google Scholar

[24] T. D. Wooley, On Waring’s problem for intermediate powers, Acta Arith. 176 (2016), no. 3, 241–247. 10.4064/aa8439-8-2016Search in Google Scholar

Received: 2022-12-02
Revised: 2023-09-18
Published Online: 2023-11-01
Published in Print: 2023-12-01

© 2023 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 13.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/crelle-2023-0072/html
Scroll to top button