Available in electronic format
Available in print format		 Transacrions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(e) ISSN 0002-9947(p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Previous Article
     

Upper bounds for the number of solutions of a Diophantine equation

Author(s): M. Z. Garaev
Journal: Trans. Amer. Math. Soc. 357 (2005), 2527-2534.
MSC (2000): Primary 11D45, 11L03
Posted: December 28, 2004
Retrieve article in: PDF DVI PostScript

Abstract | References | Similar articles | Additional information

Abstract: We give upper bound estimates for the number of solutions of a certain diophantine equation. Our results can be applied to obtain new lower bound estimates for the $L_1$-norm of certain exponential sums.

References:

1.
S. V. Bochkarev, A method for estimating the $L_1$-norm of an exponential sum, Proc. Steklov Inst. Math., 218, 74-78 (1997). MR 99h:11093

2.
G. Elekes, M. B. Nathanson and I. Z. Ruzsa, Convexity and Sumsets, J. Number Theory, 83, 194-201 (2000). MR 2001e:11020

3.
M. Z. Garaev, On lower bounds for the $L_1$-norm of exponential sums, Math. Notes, 68, 713-720 (2000). MR 2002f:11109

4.
M. Z. Garaev, On the Waring-Goldbach problem with small non-integer exponent, Acta Arith. 108.3 (2003), 297-302. MR 2004c:11183

5.
M. Z. Garaev and Ka-Lam Kueh, $L_1$-norms of exponential sums and the corresponding additive problem, Z. Anal. Anwendungen, 20, 999-1006 (2001).MR 2002k:11141

6.
A. A. Karatsuba, An estimate of the $L_1$-norm of an exponential sum, Math. Notes, 64, 401-404 (1998). MR 2000a:11113

7.
S. V. Konyagin, On the problem of Littlewood, Izv. Acad. Nauk SSSR Ser. Mat. [Math. USSR-Izv.], 45(2), 243-265 (1981). MR 83d:10045

8.
S. V. Konyagin, An estimate of the $L_1$-norm of an exponential sum, The Theory of Approximations of Functions and Operators. Abstracts of Papers of the International Conference Dedicated to Stechkin's 80th Anniversary [in Russian]. Ekaterinburg (2000), pp. 88-89.

9.
O. C. McGehee, L. Pigno and B. Smith, Hardy's inequality and the $L_1$ norm of exponential sums, Ann. of Math.(2), 113(3), 613-618 (1981). MR 83c:43002b

10.
I. I. Piatetski-Shapiro, On a variant of Waring-Goldbach's problem, Mat. Sb. 30 (1952), 105-120 (in Russian). MR 14:451d

11.
A. Zygmund, Trigonometric Series, Vol. 1. Cambridge Univ. Press, Cambridge (1959). MR 21:6498

Similar Articles:

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 11D45, 11L03

Retrieve articles in all Journals with MSC (2000): 11D45, 11L03

Additional Information:

M. Z. Garaev
Affiliation: Instituto de Matemáticas UNAM, Campus Morelia, Ap. Postal 61-3 (Xangari) CP 58089, Morelia, Michoacán, México
Email: garaev@matmor.unam.mx

DOI: 10.1090/S0002-9947-04-03611-6
PII: S 0002-9947(04)03611-6
Keywords: Diophantine equation, Karatsuba's theorem, Konyagin's estimate
Received by editor(s): September 8, 2003
Received by editor(s) in revised form: January 10, 2004
Posted: December 28, 2004
Copyright of article: Copyright 2004, American Mathematical Society

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2008, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google