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Double exponential sums with exponential functions. (English) Zbl 1428.11143

Summary: We obtain several estimates for double rational exponential sums modulo a prime \(p\) with products \(n g^m\) where both \(n\) and \(m\) run through short intervals and \(g\) is fixed integer. We also obtain some new estimates for the number of points on exponential modular curves \(a g^m \equiv n \pmod {p}\) and similar.

MSC:

11L07 Estimates on exponential sums
11D79 Congruences in many variables
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[1] E. Aksoy Yazici, B. Murphy, M. Rudnev and I. D. Shkredov, Growth estimates in positive characteristic via collisions, to appear in Int. Math. Res. Not. · Zbl 1405.11008
[2] Ayyad, A., Cochrane, T. and Zheng, Z., The congruence \(x_1 x_2 \equiv x_3 x_4(\text{mod} p)\), the equation \(x_1 x_2 = x_3 x_4\) and mean values of character sums, J. Number Theory59 (1996) 398-413. · Zbl 0869.11003
[3] W. D. Banks and I. E. Shparlinski. Bounds on short character sums and \(L\)-functions for characters with a smooth modulus, preprint (2016); arXiv:1605.07553, http://arxiv.org/abs/1605.07553.
[4] Bourgain, J., On the distribution of the residues of small multiplicative subgroups of \(\mathbb{F}_p\), Israel J. Math.172 (2009) 61-74. · Zbl 1197.11015
[5] Bourgain, J., Garaev, M. Z., Konyagin, S. V. and Shparlinski, I. E., On congruences with products of variables from short intervals and applications, Proc. Steklov Math. Inst.280 (2013) 67-96. · Zbl 1301.11041
[6] Bourgain, J., Garaev, M. Z., Konyagin, S. V. and Shparlinski, I. E., Multiplicative congruences with variables from short intervals, J. d’Anal. Math.124 (2014) 117-147. · Zbl 1385.11002
[7] Chan, T. H. and Shparlinski, I. E., On the concentration of points on modular hyperbolas and exponential curves, Acta Arith.142 (2010) 59-66. · Zbl 1198.11002
[8] Cilleruelo, J. and Garaev, M. Z., Concentration of points on two and three dimensional modular hyperbolas and applications, Geom. Funct. Anal.21 (2011) 892-904. · Zbl 1225.11004
[9] Hegyvári, N. and Hennecart, F., Distribution of residues in approximate subgroups of \(\mathbb{F}_p^\ast \), Proc. Amer. Math. Soc.140 (2012) 1-6. · Zbl 1262.11042
[10] Iwaniec, H. and Kowalski, E., Analytic Number Theory (American Mathematical Society, Providence, RI, 2004). · Zbl 1059.11001
[11] Kerr, B., Incomplete exponential sums over exponential functions, Oxford J. Math.66 (2015) 213-224. · Zbl 1386.11095
[12] Konyagin, S. V. and Shparlinski, I. E., On the consecutive powers of a primitive root: Gaps and exponential sums, Mathematika58 (2012) 11-20. · Zbl 1276.11005
[13] Korobov, N. M., On the distribution of digits in periodic fractions, Matem. Sbornik89 (1972) 654-670 (in Russian). · Zbl 0248.10007
[14] Roche-Newton, O., Rudnev, M. and Shkredov, I. D., New sum-product type estimates over finite fields, Adv. Math.293 (2016) 589-605. · Zbl 1412.11018
[15] Shkredov, I. D., Some new inequalities in additive combinatorics, Moscow J. Comb. Number Theory3 (2013) 425-475. · Zbl 1382.11017
[16] Shkredov, I. D., On exponential sums over multiplicative subgroups of medium size, Finite Fields Appl.30 (2014) 72-87. · Zbl 1300.11085
[17] Shparlinski, I. E., Distribution of exponential functions modulo a prime power, J. Number Theory143 (2014) 224-231. · Zbl 1356.11002
[18] Shparlinski, I. E., On small gaps between the elements of multiplicative subgroups of finite fields, Des. Codes Cryptogr.80 (2016) 63-71. · Zbl 1367.11040
[19] Shparlinski, I. E. and Yau, K.-h., Bounds of double multiplicative character sums and gaps between residues of exponential functions, J. Number Theory167 (2016) 304-316. · Zbl 1419.11004
[20] I. E. Shparlinski and T. P. Zhang, Cancellations amongst Kloosterman sums, to appear in Acta Arith.. · Zbl 1368.11092
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