Hostname: page-component-848d4c4894-75dct Total loading time: 0 Render date: 2024-05-19T09:07:45.328Z Has data issue: false hasContentIssue false
  • English
  • Français

WARING’S PROBLEM WITH PIATETSKI-SHAPIRO NUMBERS

Part of: Sequences and sets Exponential sums and character sums Additive number theory; partitions

Published online by Cambridge University Press:  17 February 2016

Yıldırım Akbal  and
Ahmet M. Güloğlu
Yıldırım Akbal
Affiliation:
Department of Mathematics, Bilkent University, 06800 Bilkent, Ankara, Turkey email yildirim.akbal@bilkent.edu.tr
Ahmet M. Güloğlu
Affiliation:
Department of Mathematics, Bilkent University, 06800 Bilkent, Ankara, Turkey email guloglua@fen.bilkent.edu.tr
Get access

Abstract

In this paper, we investigate in various ways the representation of a large natural number as a sum of a fixed power of Piatetski-Shapiro numbers, thereby establishing a variant of the Hilbert–Waring problem with numbers from a sparse sequence.

MSC classification

Primary: 11P05: Waring's problem and variants
Secondary: 11P55: Applications of the Hardy-Littlewood method 11L07: Estimates on exponential sums 11L15: Weyl sums 11B83: Special sequences and polynomials
Type
Research Article
Information
Mathematika , Volume 62 , Issue 2 , 2016 , pp. 524 - 550
Copyright
Copyright © University College London 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Arkhipov, G. I. and Zhitkov, A. N., Waring’s problem with nonintegral exponents. Izv. Akad. Nauk SSSR Ser. Mat. 48(6) 1984, 1138–1150 ; English translation: Math. USSR-Izv. 25(3) (1985), 443–454.Google Scholar
Davenport, H., On Waring’s problem for fourth powers. Ann. of Math. (2) 40 1939, 731–747.Google Scholar
Deshouillers, J. M., Problème de Waring avec exposants non entiers. Bull. Soc. Math. France 101 1973, 285–295.Google Scholar
Graham, S. W. and Kolesnik, G., Van der Corput’s Method of Exponential Sums (London Mathematical Society Lecture Note Series 126 ), Cambridge University Press (Cambridge, 1991).Google Scholar
Halberstam, H. and Richert, H. E., Sieve Methods, Academic Press (London, 1974).Google Scholar
Hilbert, D., Beweis für die Darstellbarkeit der ganzen Zahlen durch eine feste Anzahl n ter Potenzen (Waringsches Problem). Math. Ann. 67(3) 1909, 281–300.Google Scholar
Iwaniec, H. and Kowalski, E., Analytic Number Theory (American Mathematical Society Colloquium Publications 53 ), American Mathematical Society (Providence, RI, 2004).Google Scholar
Listratenko, Y. R., Estimation of the number of summands in Waring’s problem for the integer parts of fixed powers of nonnegative numbers. Chebyshevkiĭ Sb. 3(2(4)) 2002, 71–77 (in Russian).Google Scholar
Piatetski-Shapiro, I. I., On the distribution of prime numbers in sequences of the form [f (n)]. Mat. Sb. N.S. 33(75) 1953, 559–566 (Russian).Google Scholar
Segal, B. I., On a theorem analogous to Waring’s theorem. Dokl. Akad. Nauk SSSR 1 1933, 47–49 (Russian).Google Scholar
Segal, B. I., Waring’s theorem for powers with fractional and irrational exponents. Tr. Fiz.-Mat. Inst. Steklov. Otdel. Mat. 5 1934, 73–86 (Russian).Google Scholar
Vaughan, R. C., The Hardy–Littlewood Method (Cambridge Tracts in Mathematics 125 ), 2nd edn., Cambridge University Press (Cambridge, 1997).Google Scholar
Vaughan, R. C., A new iterative method in Waring’s problem. Acta Math. 162(1–2) 1989, 1–71.Google Scholar
Waring, E., Meditationes Algebraicae (translation by Dennis Weeks of the 1782 edition), 2nd edn., American Mathematical Society (Providence, RI, 1982).Google Scholar
Whittaker, E. T. and Watson, G. N., Modern Analysis, Cambridge University Press (Cambridge, 1927).Google Scholar
Wooley, T. D., New estimates for smooth Weyl sums. J. Lond. Math. Soc. (2) 51(1) 1995, 1–13.CrossRefGoogle Scholar
Wooley, T. D., The asymptotic formula in Waring’s problem. Int. Math. Res. Not. IMRN 2012(7) 2012, 1485–1504.Google Scholar
Wooley, T. D., Multigrade efficient congruencing and Vinogradov’s mean value theorem. Proc. Lond. Math. Soc. (3) 111(3) 2015, 519–560.Google Scholar
Wooley, T. D., Vinogradov’s mean value theorem via efficient congruencing, II. Duke Math. J. 162(4) 2013, 673–730.Google Scholar
Wooley, T. D., The cubic case of the main conjecture in Vinogradov’s mean value theorem, Preprint, 2014, arXiv:1401.3150 [math.NT].Google Scholar