Bilinear sums with exponential functions

Shparlinski, Igor

doi:10.1090/S0002-9939-09-09882-7

Bilinear sums with exponential functions
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by Igor E. Shparlinski PDF

Proc. Amer. Math. Soc. 137 (2009), 2217-2224 Request permission

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 $M$ and $N$, we estimate the exponential sums \[ \sum _{\substack {p \le M \mathrm {gcd}(ag,p) =1}} \sum _{1 \le n \le N} \varphi _p\psi _n \mathbf {e}_p\left (a g^n\right ), \qquad a \in \mathbb {Z}, \] where the outer summation is taken over all primes $p \le M$ with $\mathrm {gcd}(ag,p) =1$.

References

William D. Banks, Alessandro Conflitti, John B. Friedlander, and Igor E. Shparlinski, Exponential sums over Mersenne numbers, Compos. Math. 140 (2004), no. 1, 15–30. MR 2004121, DOI 10.1112/S0010437X03000022
William D. Banks, John B. Friedlander, Moubariz Z. Garaev, and Igor E. Shparlinski, Character sums with exponential functions over smooth numbers, Indag. Math. (N.S.) 17 (2006), no. 2, 157–168. MR 2321378, DOI 10.1016/S0019-3577(06)80013-3
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).
J. Bourgain, Estimates on exponential sums related to the Diffie-Hellman distributions, Geom. Funct. Anal. 15 (2005), no. 1, 1–34. MR 2140627, DOI 10.1007/s00039-005-0500-4
J. Bourgain, Exponential sum estimates over subgroups of ${\Bbb Z}^*_q$, $q$ arbitrary, J. Anal. Math. 97 (2005), 317–355. MR 2274981, DOI 10.1007/BF02807410
J. Bourgain, Exponential sum estimates in finite commutative rings and applications, J. Anal. Math. 101 (2007), 325–355. MR 2346549, DOI 10.1007/s11854-007-0012-2
J. Bourgain and M.-C. Chang, Exponential sum estimates over subgroups and almost subgroups of $\Bbb Z_Q^*$, where $Q$ is composite with few prime factors, Geom. Funct. Anal. 16 (2006), no. 2, 327–366. MR 2231466, DOI 10.1007/s00039-006-0558-7
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.
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. London Math. Soc. (2) 73 (2006), no. 2, 380–398. MR 2225493, DOI 10.1112/S0024610706022721
Michael Dewar, Daniel Panario, and Igor E. Shparlinski, Distribution of exponential functions with $k$-full exponent modulo a prime, Indag. Math. (N.S.) 15 (2004), no. 4, 497–503. MR 2114933, DOI 10.1016/S0019-3577(04)80014-4
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).
M. Z. Garaev and I. E. Shparlinski, The large sieve inequality with exponential functions and the distribution of Mersenne numbers modulo primes, Int. Math. Res. Not. 39 (2005), 2391–2408. MR 2181356, DOI 10.1155/IMRN.2005.2391
Sergei V. Konyagin and Igor E. Shparlinski, Character sums with exponential functions and their applications, Cambridge Tracts in Mathematics, vol. 136, Cambridge University Press, Cambridge, 1999. MR 1725241, DOI 10.1017/CBO9780511542930
N. M. Korobov, The distribution of digits in periodic fractions, Mat. Sb. (N.S.) 89(131) (1972), 654–670, 672 (Russian). MR 0424660
A. G. Postnikov, Ergodic aspects of the theory of congruences and of the theory of Diophantine approximations, Trudy Mat. Inst. Steklov 82 (1966), 3–112 (Russian). MR 0214561
Igor Shparlinski, Cryptographic applications of analytic number theory, Progress in Computer Science and Applied Logic, vol. 22, Birkhäuser Verlag, Basel, 2003. Complexity lower bounds and pseudorandomness. MR 1954519, DOI 10.1007/978-3-0348-8037-4
Igor E. Shparlinski, Distribution of exponential functions with squarefull exponent in residue rings, Indag. Math. (N.S.) 15 (2004), no. 2, 283–289. MR 2071861, DOI 10.1016/S0019-3577(04)90020-1
Alev Topuzoğlu and Arne Winterhof, Pseudorandom sequences, Topics in geometry, coding theory and cryptography, Algebr. Appl., vol. 6, Springer, Dordrecht, 2007, pp. 135–166. MR 2278037, DOI 10.1007/1-4020-5334-4_{4}