Proceedings of the American Mathematical Society

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ISSN 1088-6826(e) ISSN 0002-9939(p)

Bilinear sums with exponential functions

Author(s): Igor E. Shparlinski
Journal: Proc. Amer. Math. Soc. 137 (2009), 2217-2224.
MSC (2000): Primary 11L07, 11L26
Posted: March 4, 2009
MathSciNet review: 2495254
Retrieve article in: PDF

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Abstract: Let $g\ne 0, \pm 1$ be a fixed integer. Given two sequences of complex numbers $\left(\varphi_m\right)_{m=1}^\infty$ and $\left(\psi_n\right)_{n=1}^\infty$ and two sufficiently large integers and , we estimate the exponential sums

$\displaystyle \sum_{\substack{p \le M \mathrm{gcd}(ag,p) =1}} \sum_{1 \le n \le N} \varphi_p\psi_n \mathbf{e}_p\left(a g^n\right), \qquad a \in \mathbb{Z},$

where the outer summation is taken over all primes $p \le M$ with $\mathrm{gcd}(ag,p) =1$ .

References:

1.: W. Banks, A. Conflitti, J. B. Friedlander and I. E. Shparlinski, `Exponential sums over Mersenne numbers', Compos. Math., 140 (2004), 15-30. MR 2004121 (2004j:11091)
2.: W. Banks, J. B. Friedlander, M. Z. Garaev and I. E. Shparlinski, `Character sums with exponential functions over smooth numbers', Indag. Math., 17 (2006), 157-168. MR 2321378 (2008e:11097)
3.: W. D. Banks, M. Z. Garaev, F. Luca and I. E. Shparlinski, `Uniform distribution of fractional parts related to pseudoprimes', Canad. J. Math. (to appear).
4.: J. Bourgain, `Estimates on exponential sums related the Diffie-Hellman distributions', Geom. Funct. Anal., 15 (2005), 1-34. MR 2140627 (2006h:11095)
5.: J. Bourgain, `Exponential sum estimates over subgroups of $\mathbb{Z}_q^\ast$ , arbitrary', J. Anal. Math., 97 (2005), 317-355. MR 2274981 (2007j:11103)
6.: J. Bourgain, `Exponential sum estimates in finite commutative rings and applications', J. Anal. Math., 101 (2007), 325-355. MR 2346549 (2008i:11108)
7.: J. Bourgain and M. Chang, `Exponential sum estimates over subgroups and almost subgroups of $\mathbb{Z}_Q^\ast$ , where is composite with few prime factors', Geom. Funct. Anal., 16 (2006), 327-366. MR 2231466 (2007d:11093)
8.: J. Bourgain and M. Z. Garaev, `On a variant of sum-product estimates and explicit exponential sum bounds in prime fields', Math. Proc. Cambr. Phil. Soc., 146 (2008), 1-21.
9.: J. Bourgain, A. A. Glibichuk and S. V. Konyagin, `Estimates for the number of sums and products and for exponential sums in fields of prime order', J. Lond. Math. Soc., 73 (2006), 380-398. MR 2225493 (2007e:11092)
10.: M. Dewar, D. Panario and I. E. Shparlinski, `Distribution of exponential functions with -full exponent modulo a prime', Indag. Math., 15 (2004), 497-503. MR 2114933 (2005k:11166)
11.: M. Z. Garaev, `The large sieve inequality for the exponential sequence $\lambda^{[O(n^{15/14+o(1)})]}$ modulo primes', Canad. J. Math. (to appear).
12.: M. Z. Garaev and I. E. Shparlinski, `The large sieve inequality with exponential functions and the distribution of Mersenne numbers modulo primes', Intern. Math. Res. Notices, 2005:39 (2005), 2391-2408. MR 2181356 (2006i:11108)
13.: S. V. Konyagin and I. E. Shparlinski, Character sums with exponential functions and their applications, Cambridge Univ. Press, Cambridge, 1999. MR 1725241 (2000h:11089)
14.: N. M. Korobov, `On the distribution of digits in periodic fractions', Matem. Sbornik, 89(131) (1972), 654-670 (in Russian). MR 0424660 (54:12619)
15.: A. G. Postnikov, Ergodic aspects of the theory of congruences and of the theory of Diophantine approximations, Trudy Mat. Inst. Steklov, vol. 82, 1966 (Russian); translated by the Amer. Math. Soc., Providence, R.I., 1967. MR 0214561 (35:5410)
16.: I. E. Shparlinski, Cryptographic applications of analytic number theory, Birkhäuser Verlag, Basel, 2003. MR 1954519 (2004h:94049)
17.: I. E. Shparlinski, `Distribution of exponential functions with squarefull exponent in residue rings', Indag. Math., 15 (2004), 283-289. MR 2071861 (2005h:11183)
18.: A. Topuzoglu and A. Winterhof, `Pseudorandom sequences', Topics in Geometry, Coding Theory and Cryptography, Springer, Dordrecht, 2007, 135-166. MR 2278037 (2007m:11106)

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