Waring’s problem for Beatty sequences and a local to global principle

Banks, William D.; Güloğlu, Ahmet M.; Vaughan, Robert C.

Journal de théorie des nombres de Bordeaux, Tome 26 (2014) no. 1, pp. 1-16.

Nous examinons de façons diverses la représentation d’un grand nombre entier $N$ comme somme de $s$ entiers positifs qui sont tous des puissances $k$-ième de termes d’une suite de Beatty donnée. Entre autres, une forme très générale du principe local-global est établie dans la théorie additive des nombres. La démonstration est courte mais elle utilise un théorème profond de M. Kneser.

We investigate in various ways the representation of a large natural number $N$ as a sum of $s$ positive $k$-th powers of numbers from a fixed Beatty sequence. Inter alia, a very general form of the local to global principle is established in additive number theory. Although the proof is very short, it depends on a deep theorem of M. Kneser.

Classification : 11P05

[1] K. D. Boklan, The asymptotic formula in Waring’s problem. Mathematika 41 (1994), no. 2, 329–347. | MR | Zbl

[2] H. Davenport, Analytic methods for Diophantine equations and Diophantine inequalities. Second edition. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 2005. | MR | Zbl

[3] H. Halberstam and H.-E. Richert, Sieve methods. London Mathematical Society Monographs, No. 4. Academic Press, London-New York, 1974. | MR | Zbl

[4] H. Halberstam and K. F. Roth, Sequences. Second edition. Springer-Verlag, New York-Berlin, 1983. | MR | Zbl

[5] D. Hilbert, Beweis für die Darstellbarkeit der ganzen Zahlen durch eine feste Anzahl $n^{ter}$ Potenzen (Waringsches Problem). Math. Ann. 67 (1909), no. 3, 281–300. | EuDML | JFM | MR

[6] H. L. Montgomery and R. C. Vaughan, Multiplicative number theory. I. Classical theory. Cambridge Studies in Advanced Mathematics, 97. Cambridge University Press, Cambridge, 2007. | MR | Zbl

[7] E. C. Titchmarsh, The theory of the Riemann zeta-function. Second edition. The Clarendon Press, Oxford University Press, New York, 1986. | Zbl

[8] R. C. Vaughan, On Waring’s problem for cubes. J. Reine Angew. Math. 365 (1986), 122–170. | EuDML | MR | Zbl

[9] R. C. Vaughan, On Waring’s problem for smaller exponents II. Mathematika 33 (1986), no. 1, 6–22. | MR | Zbl

[10] R. C. Vaughan, A new iterative method in Waring’s problem. Acta Math. 162 (1989), no. 1-2, 1–71. | Zbl

[11] R. C. Vaughan, The Hardy-Littlewood method. Second edition. Cambridge Tracts in Mathematics, 125. Cambridge University Press, Cambridge, 1997. | MR | Zbl

[12] R. C. Vaughan, On generating functions in additive number theory, I, in Analytic Number Theory, Essays in Honour of Klaus Roth, 436–448. Cambridge Univ. Press, Cambridge, 2009. | Zbl

[13] R. C. Vaughan and T. D. Wooley, Further improvements in Waring’s problem, III: Eighth powers. Philos. Trans. Roy. Soc. London Ser. A 345 (1993), no. 1676, 385–396. | MR | Zbl

[14] 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. | MR | Zbl

[15] R. C. Vaughan and T. D. Wooley, Further improvements in Waring’s problem. Acta Math. 174 (1995), no. 2, 147–240. | MR | Zbl

[16] R. C. Vaughan and T. D. Wooley, Further improvements in Waring’s problem, IV: Higher powers. Acta Arith. 94 (2000), no. 3, 203–285. | MR | Zbl

[17] E. Waring, Meditationes algebraicæ. Cambridge, England, 1770.

[18] T. D. Wooley, Vinogradov’s mean value theorem via efficient congruencing. Ann. Math., to appear. | MR | Zbl