Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-30T20:17:28.656Z Has data issue: false hasContentIssue false
  • English
  • Français

ON SOME MULTIPLE CHARACTER SUMS

Part of: Finite fields and commutative rings (number-theoretic aspects) Sequences and sets Exponential sums and character sums

Published online by Cambridge University Press:  03 April 2017

Ilya D. Shkredov  and
Igor E. Shparlinski
Ilya D. Shkredov
Affiliation:
Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow 119991, Russia Institute for Information Transmission Problems of Russian Academy of Sciences, Bolshoy Karetny Per. 19, Moscow 127994, Russia email ilya.shkredov@gmail.com
Igor E. Shparlinski
Affiliation:
Department of Pure Mathematics, University of New South Wales, Sydney, NSW 2052, Australia email igor.shparlinski@unsw.edu.au
Get access

Abstract

We improve a recent result of B. Hanson [Estimates for character sums with various convolutions. Preprint, 2015, arXiv:1509.04354] on multiplicative character sums with expressions of the type $a+b+cd$ and variables $a,b,c,d$ from four distinct sets of a finite field. We also consider similar sums with $a+b(c+d)$. Our new bounds rely on some recent advances in additive combinatorics.

MSC classification

Primary: 11B30: Arithmetic combinatorics; higher degree uniformity 11L07: Estimates on exponential sums 11T23: Exponential sums
Type
Research Article
Information
Mathematika , Volume 63 , Issue 2 , 2017 , pp. 553 - 560
Copyright
Copyright © University College London 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aksoy Yazici, E., Murphy, B., Rudnev, M. and Shkredov, I. D., Growth estimates in positive characteristic via collisions. Int. Math. Res. Not. IMRN (2016), doi:10.1093/imrn/rnw206.Google Scholar
Balog, A. and Wooley, T. D., A low-energy decomposition theorem. Q. J. Math. (to appear).Google Scholar
Bourgain, J., Multilinear exponential sums in prime fields under optimal entropy condition on the sources. Geom. Funct. Anal. 18 2009, 1477–1502.CrossRefGoogle Scholar
Bourgain, J., On Exponential Sums in Finite Fields (Bolyai Society Mathematical Studies 21 ), János Bolyai Mathematical Society (Budapest, 2010), 219–242.Google Scholar
Bourgain, J. and Garaev, M. Z., On a variant of sum–product estimates and explicit exponential sum bounds in prime fields. Math. Proc. Cambridge Philos. Soc. 146 2009, 1–21.CrossRefGoogle Scholar
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 Inst. Math. 280 2013, 61–90 (transl. from Tr. Mat. Inst. Steklova).Google Scholar
Bourgain, J. and Glibichuk, A., Exponential sum estimates over a subgroup in an arbitrary finite field. J. Anal. Math. 115 2011, 51–70.CrossRefGoogle Scholar
Chang, M.-C., On a question of Davenport and Lewis and new character sum bounds in finite fields. Duke Math. J. 145 2008, 409–442.Google Scholar
Davenport, H. and Erdős, P., The distribution of quadratic and higher residues. Publ. Math. Debrecen 2 1952, 252–265.CrossRefGoogle Scholar
Friedlander, J. B. and Iwaniec, H., Estimates for character sums. Proc. Amer. Math. Soc. 119 1993, 365–372.Google Scholar
Garaev, M. Z., Sums and products of sets and estimates of rational trigonometric sums in fields of prime order. Russian Math. Surveys 65 2010, 599–658 (transl. from Uspekhi Mat. Nauk).Google Scholar
Hanson, B., Estimates for character sums with various convolutions. Preprint, 2015,arXiv:1509.04354.Google Scholar
Iwaniec, H. and Kowalski, E., Analytic Number Theory, American Mathematical Society (Providence, RI, 2004).Google Scholar
Karatsuba, A. A., The distribution of values of Dirichlet characters on additive sequences. Dokl. Acad. Sci. USSR 319 1991, 543–545 (in Russian).Google Scholar
Konyagin, S. V., Estimates for character sums in finite fields. Math. Notes 88 2010, 503–515 (transl. from Mat. Zametki).Google Scholar
Ostafe, A., Polynomial values in affine subspaces over finite fields. J. Anal. Math. (to appear).Google Scholar
Petridis, G. and Shparlinski, I. E., Bounds on trilinear and quadrilinear exponential sums. J. Anal. Math. (to appear).Google Scholar
Roche-Newton, O., Rudnev, M. and Shkredov, I. D., New sum–product type estimates over finite fields. Adv. Math. 293 2016, 589–605.Google Scholar
Rudnev, M., On the number of incidences between planes and points in three dimensions. Combinatorica (2017), doi:10.1007/s00493-016-3329-6.Google Scholar
Shkredov, I. D. and Volostnov, A. S., Sums of multiplicative characters with additive convolutions. Proc. Steklov Math. Inst. 296 2017, (to appear).Google Scholar