Upper bounds for the number of solutions of a Diophantine equation

Author:
M. Z. Garaev

Journal:
Trans. Amer. Math. Soc. **357** (2005), 2527-2534

MSC (2000):
Primary 11D45, 11L03

DOI:
https://doi.org/10.1090/S0002-9947-04-03611-6

Published electronically:
December 28, 2004

MathSciNet review:
2140449

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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 -norm of certain exponential sums.

**1.**S. V. Bochkarev,*A method for estimating the**-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**-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,*-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**-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**-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**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**

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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:
https://doi.org/10.1090/S0002-9947-04-03611-6

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

Published electronically:
December 28, 2004

Article copyright:
© Copyright 2004
American Mathematical Society