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Multiplicative Energy of Shifted Subgroups and Bounds On Exponential Sums with Trinomials in Finite Fields

Published online by Cambridge University Press:  20 November 2018

Simon Macourt ,
Ilya D. Shkredov  and
Igor E. Shparlinski
Simon Macourt
Affiliation:
Department of Pure Mathematics, University of New South Wales, Sydney, NSW 2052, Australia, e-mail: [email protected]
Ilya D. Shkredov
Affiliation:
Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow, Russia, 119991, and Institute for Information Transmission Problems of Russian Academy of Sciences, Bolshoy Karetny Per. 19, Moscow, Russia, 127994, and MIPT, Institutskii per. 9, Dolgoprudnii, Russia, 141701, e-mail: [email protected]
Igor E. Shparlinski
Affiliation:
Department of Pure Mathematics, University of New South Wales, Sydney, NSW 2052, Australia, e-mail: [email protected]
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Abstract

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We prove a new bound on collinear triples in subgroups of prime finite fields and use it to give some new bounds on exponential sums with trinomials.

Keywords

11L0711T23exponential sumsparse polynomialtrinomial
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 70 , Issue 6 , 01 December 2018 , pp. 1319 - 1338
Copyright
Copyright © Canadian Mathematical Society 2018

References

[1] Akulinichev, N. M., Estimates for rational trigonometric sums of a special type. (Russian) Dokl. Akad. Nauk SSSR 161 (1965), 743–745.Google Scholar
[2] Balog, A., Another sum-product estimate in finite fields. Proc. Steklov Inst. Math. 280 (2013), Suppl. 2, S23-S29. http://dx.doi.Org/10.1134/S0081 543813030024Google Scholar
[3] Bourgain, J., Multilinear exponential sums in prime fields under optimal entropy condition on the sources. Geom. and Funct. Anal. 18 (2009), 1477–1502. http://dx.doi.org/10.1007/s00039-008-0691-6Google Scholar
[4] Bourgain, J., Estimates of polynomial exponential sums. Israel J. Math. 176 (2010), 221–240. http://dx.doi.org/10.1007/s11856-010-0027-8Google Scholar
[5] Bourgain, J. and Glibichuk, A., Exponential sum estimates over a subgroup in an arbitrary finite field. J. Anal. Math. 115 (2011), 51–70. http://dx.doi.Org/10.1007/s11854-011-0023-xGoogle Scholar
[6] Bourgain, J., Glibichuk, A. A., and Konyagin, S. V., Estimates for the number of sums and products and for exponential sums infields of prime order. J. London Math. Soc. 73 (2006), 380–398. http://dx.doi.org/10.1112/S0024610706022721Google Scholar
[7] Bourgain, J., Katz, N. H., and Tao, T., A sum-product estimate in finite fields, and applications. Geom. Funct. Anal. 14 (2004), 27–57. http://dx.doi.org/10.1007/s00039-004-0451-1Google Scholar
[8] Cochrane, T. T., Coffelt, J., and Pinner, C. G., A further refinement ofMordell's bound on exponential sums. Acta Arith. 116 (2005), 35–41. http://dx.doi.Org/10.4064/aa116-1-4Google Scholar
[9] Cochrane, T. T., Coffelt, J., and Pinner, C. G., A system of simultaneous congruences arising from trinomial exponential sums. J. Théor. Nombres Bordeaux 18 (2006), 59–72. http://dx.doi.Org/10.58O2/jtnb.533Google Scholar
[10] Cochrane, T. and Pinner, C., An improved Mordell type bound for exponential sums. Proc. Amer. Math. Soc. 133 (2005), 313–320. http://dx.doi.org/10.1090/S0002-9939-04-07726-3Google Scholar
[11] Cochrane, T. and Pinner, C., Using Stepanov's method for exponential sums involving rational functions. J. Number Theory 116 (2006), 270–292. http://dx.doi.Org/10.1016/j.jnt.2005.04.001Google Scholar
[12] Cochrane, T. and Pinner, C., Bounds on fewnomial exponential sums over Zp. Math. Proc. Cambridge Philos. Soc. 149 (2010), 217–227.Google Scholar
[13] Cochrane, T. and Pinner, C., Explicit bounds on monomial and binomial exponential sums. Q. J. Math. 62 (2011), 323–349. http://dx.doi.org/10.1017/S0305004110000319Google Scholar
[14] Crocker, R., On the sum of a prime and of two powers of two. Pacific J. Math. 36 (1971), 103–107. http://dx.doi.Org/10.2140/pjm.1971.36.103Google Scholar
[15] Erdôs, P. and Murty, R., On the order of a (mod p). CRM Proc. Lecture Notes, American Mathematical Society, Providence, RI, 1999, pp. 87-97.Google Scholar
[16] Ford, K., The distribution of integers with a divisor in a given interval. Ann. of Math. 168 (2008), 367–433. http://dx.doi.Org/10.4007/annals.2008.168.367Google Scholar
[17] Garaev, M. Z., Sums and products of sets and estimates of rational trigonometric sums infields of prime order. Russian Math. Surveys 65 (2010), no. 4, 599–658. http://dx.doi.Org/10.4213/rm9367Google Scholar
[18] Glibichuk, A., Combinational properties of sets of residues modulo a prime and the Erdos-Graham problem. Math. Notes 79 (2006), 356–365. http://dx.doi.org/10.4213/mzm2708Google Scholar
[19] Heath-Brown, D. R. and Konyagin, S. V., New bounds for Gauss sums derived from kth powers, and for Heilbronn's exponential sum. Q. J. Math. 51 (2000), 221–235. http://dx.doi.Org/10.1093/qjmath/51.2.221Google Scholar
[20] Indlekofer, H.-K. and Timofeev, N. M., Divisors of shifted primes. Publ. Math. Debrecen 60 (2002), 307–345.Google Scholar
[21] Konyagin, S. V., Bounds of exponential sums over subgroups and Gauss sums. (Russian), Proc. 4th Intern. Conf. Modern Problems of Number Theory and Its Applications, Moscow Lomonosov State Univ., Moscow, 2002, pp. 86–114.Google Scholar
[22] Macourt, S., Incidence results and bounds oftrilinear and quadrilinear exponential sums. arxiv:1 707.08268Google Scholar
[23] Mit'kin, D. A., Estimation of the total number of the rational points on a set of curves in a simple finite field. (Russian) Chebyshevsky Sbornik 4 (2003), no.4, 94-102.Google Scholar
[24] Murphy, B., Petridis, G., Roche-Newton, O., Rudnev, M. and Shkredov, I. D., New results on sum-product type growth in positive characteristic. arxiv:1702.01003Google Scholar
[25] Petridis, G. and Shparlinski, I. E., Bounds on trilinear and quadrilinear exponential sums. J. d'Analyse Math., to appear.Google Scholar
[26] Romanoff, N. P., fiber einige Sàtze der additiven Zahlentheorie. Math. Ann. 109 (1934), 668–678. http://dx.doi.Org/10.1007/BF014491 61Google Scholar
[27] Shkredov, I. D., On exponential sums over multiplicative subgroups of medium size. Finite Fields Appl. 30 (2014), 72–87. http://dx.doi.Org/10.1016/j.ffa.2O14.06.002Google Scholar
[28] Shkredov, I. D., On tripling constant of multiplicative subgroups. Integers 16 (2016), no. A75.Google Scholar
[29] Shkredov, I. D., Differences of subgroups in subgroups. Integers, to appear.Google Scholar
[30] Shkredov, I. D., Some remarks on the asymmetric sum-product phenomenon. Moscow J. Combin. and Number Theory, to appear.Google Scholar
[31] Shkredov, I. D. and Vyugin, I. V., On additive shifts of multiplicative subgroups. (Russian) Mat. Sb. 203 (2012), 81–100.Google Scholar
[32] Shparlinski, I. E., Estimates for Gaussian sums. (Russian) Mat. Zametki 50 (1991), 122–130.Google Scholar
[33] Shparlinski, I. E., On exponential sums with sparse polynomials and rational functions. J. Number Theory 60 (1996), 233–244. http://dx.doi.Org/10.1006/jnth.1996.0121Google Scholar
[34] Shteinikov, Y. N., Estimates of trigonometric sums over subgroups and some of their applications. (Russian) Mat. Zametki 98 (2015), 606–625. http://dx.doi.org/10.4213/mzm10629Google Scholar
[35] Stevens, S. and de Zeeuw, F., An improved point-line incidence bound over arbitrary fields. Bull. London Math. Soc, to appear.Google Scholar
[36] Weil, A., Basic number theory. Die Grundlehren der Mathematischen Wissenschaften, 144, Springer-Verlag, New York-Berlin, 1974.Google Scholar
[37] Tao, T. and Vu, V., Additive combinatorics. Cambridge Studies in Advanced Mathematics, 105, Cambridge University Press, Cambridge, 2006.Google Scholar
[38] Yu, H. B., Estimates for complete exponential sums of special types. Math. Proc. Cambridge Philos. Soc. 131 (2001), 321–326.Google Scholar