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On Waring’s problem: Three cubes and a sixth power

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

Abstract

We establish that almost all natural numbers not congruent to 5 modulo 9 are the sum of three cubes and a sixth power of natural numbers, and show, moreover, that the number of such representations is almost always of the expected order of magnitude. As a corollary, the number of representations of a large integer as the sum of six cubes and two sixth powers has the expected order of magnitude. Our results depend on a certain seventh moment of cubic Weyl sums restricted to minor arcs, the latest developments in the theory of exponential sums over smooth numbers, and recent technology for controlling the major arcs in the Hardy-Littlewood method, together with the use of a novel quasi-smooth set of integers.

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References

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[1] Breyer, T., Öber die Summe von sechs Kuben und zwei sechsten Potenzen, Diplomarbeit, Universität Göttingen, 1996.
[2] Brüdern, J., Iterationsmethoden in der additiven Zahlentheorie, Dissertation, Göttingen, 1988.
[3] Brüdern, J., A problem in additive number theory, Math. Proc. Cambridge Philos. Soc., 103 (1988), 2733.
[4] Brüdern, J., On Waring’s problem for cubes and biquadrates. II, Math. Proc. Cambridge Philos. Soc., 104 (1988), 199206.
[5] Brüdern, J., On Waring’s problem for cubes, Math. Proc. Cambridge Philos. Soc., 109 (1991), 229256.
[6] Brüdern, J., A note on cubic exponential sums, Sém. Théorie des Nombres, Paris, 1990–1991 (David, S., ed.), Progr. Math. 108, Birkhäuser Boston, Boston MA (1993), pp. 2334.
[7] Brüdern, J., Kawada, K. and Wooley, T. D., Additive representation in thin sequences, IV: lower bound methods, Quart. J. Math. Oxford (2) (in press); V: mixed problems of Waring type, Math. Scand. (to appear).
[8] Brüdern, J. and Wooley, T. D., On Waring’s problem for cubes and smooth Weyl sums, Proc. London Math. Soc. (3), 82 (2001), 89109.
[9] Ford, K. B., The representation of numbers as sums of unlike powers. II, J. Amer. Math. Soc., 9 (1996), 919940.
[10] Friedlander, J. B., Integers free from large and small primes, Proc. London Math. Soc. (3), 33 (1976), 565576.
[11] Hua, L.-K., On the representation of numbers as the sums of the powers of primes, Math. Z., 44 (1938), 335346.
[12] Kawada, K., On the sum of four cubes, Mathematika, 43 (1996), 323348.
[13] Lu, Ming Gao, On Waring’s problem for cubes and higher powers, Chin. Sci. Bull., 37 (1992), 14141416.
[14] Lu, Ming Gao, On Waring’s problem for cubes and fifth power, Sci. China Ser. A, 36 (1993), 641662.
[15] Saias, E., Entiers sans grand ni petit facteur premier. I, Acta Arith., 61 (1992), 347374.
[16] Tenenbaum, G., Introduction to analytic and probabilistic number theory. Cambridge Studies in Advanced Mathematics, 46, Cambridge University Press, Cambridge, 1995.
[17] Vaughan, R. C., On Waring’s problem for cubes, J. Reine Angew. Math., 365 (1986), 122170.
[18] Vaughan, R. C., On Waring’s problem for cubes II, J. London Math. Soc. (2), 39 (1989), 205218.
[19] Vaughan, R. C., A new iterative method in Waring’s problem, Acta Math., 162 (1989), 171.
[20] Vaughan, R. C., The Hardy-Littlewood method, 2nd edition, Cambridge University Press, Cambridge, 1997.
[21] Vaughan, R. C. and Wooley, T. D., On Waring’s problem: some refinements, Proc. London Math. Soc. (3), 63 (1991), 3568.
[22] Vaughan, R. C. and Wooley, T. D., Further improvements in Waring’s problem, Acta Math., 174 (1995), 147240.
[23] Wooley, T. D., Large improvements in Waring’s problem, Ann. of Math. (2), 135 (1992), 131164.
[24] Wooley, T. D. Breaking classical convexity in Waring’s problem: sums of cubes and quasi-diagonal behaviour, Inventiones Math., 122 (1995), 421451.
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On Waring’s problem: Three cubes and a sixth power

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

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