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On Waring’s problem: Three cubes and a minicube

  • Jörg Brüdern (a1) and Trevor D. Wooley (a2)

Abstract

We establish that almost all natural numbers n are the sum of four cubes of positive integers, one of which is no larger than n5/36 . The proof makes use of an estimate for a certain eighth moment of cubic exponential sums, restricted to minor arcs only, of independent interest.

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References

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[1] Boklan, K. D., A reduction technique in Waring’s problem, I, Acta Arith. 65 (1993), 147161.
[2] Brüdern, J., A problem in additive number theory, Math. Proc. Cambridge Philos. Soc. 103 (1988), 2733.
[3] Brüdern, J., Sums of four cubes, Monatsh. Math. 107 (1989), 179188.
[4] Brüdern, J., On Waring’s problem for cubes, Math. Proc. Cambridge Philos. Soc. 109 (1991), 229256.
[5] Brüdern, J. and Wooley, T. D., On Waring’s problem: three cubes and a sixth power, Nagoya Math. J. 163 (2001), 1353.
[6] Davenport, H., On Waring’s problem for cubes, Acta Math. 71 (1939), 123143.
[7] Davenport, H., Analytic Methods for Diophantine Equations and Diophantine Inequalities, 2nd ed., Cambridge University Press, Cambridge, 2005.
[8] Hunt, R. A., “On the convergence of Fourier series” in Proceedings of the Conference on Orthogonal Expansions and Their Continuous Analogues (Edwardsville, Ill., 1967), Southern Illinois University Press, Carbondale, Illinois, 1968, 235255.
[9] Kawada, K., On the sum of four cubes, Mathematika 43 (1996), 323348.
[10] Vaughan, R. C., Sums of three cubes, Bull. Lond. Math. Soc. 17 (1985), 1720.
[11] Vaughan, R. C., On Waring’s problem for cubes, J. Reine Angew. Math. 365 (1986), 122170.
[12] Vaughan, R. C., A new iterative method in Waring’s problem, Acta Math. 162 (1989), 171.
[13] Vaughan, R. C., On Waring’s problem for cubes, II, J. Lond. Math. Soc. (2) 39 (1989), 205218.
[14] Vaughan, R. C., The Hardy-Littlewood Method, 2nd ed., Cambridge University Press, Cambridge, 1997.
[15] Wooley, T. D., On simultaneous additive equations, II, J. Reine Angew. Math. 419 (1991), 141198.
[16] Wooley, T. D., Breaking classical convexity in Waring’s problem: sums of cubes and quasidiagonal behaviour, Invent. Math. 122 (1995), 421451.
[17] Wooley, T. D., Sums of three cubes, Mathematika 47 (2000), 5361.
[18] Wooley, T. D., A light-weight version of Waring’s problem, J. Austral. Math. Soc. Ser. A 76 (2004), 303316.
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On Waring’s problem: Three cubes and a minicube

  • Jörg Brüdern (a1) and Trevor D. Wooley (a2)

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