On short sums of trace functions
Annales de l'Institut Fourier, Volume 67 (2017) no. 1, pp. 423-449.

We consider sums of oscillating functions on intervals in cyclic groups of size close to the square root of the size of the group. We first prove non-trivial estimates for intervals of length slightly larger than this square root (bridging the “Polyá-Vinogradov gap” in some sense) for bounded functions with bounded Fourier transforms. We then prove that the existence of non-trivial estimates for ranges slightly below the square-root bound is stable under the discrete Fourier transform. We then give applications related to trace functions over finite fields.

Nous considérons des sommes de fonctions oscillantes sur des intervalles contenus dans un groupe fini cyclique, de taille proche de la racine carrée du cardinal du groupe. Nous démontrons tout d’abord des bornes non-triviales pour tout intervalle de longueur à peine plus grande que cette racine carrée (améliorant l’inégalité de Polyá-Vinogradov) pour les fonctions bornées dont la transformée de Fourier est bornée. Nous démontrons ensuite que l’existence d’une borne non-triviale pour un intervalle de taille un peu plus petite que la racine carrée est une propriété stable par transformation de Fourier. Nous donnons des applications liées aux fonctions trace sur les corps finis.

Published online:
DOI: 10.5802/aif.3087
Classification: 11L07, 11L05, 11T23
Keywords: Short exponential sums, trace functions, van der Corput lemma, completion method, Riemann Hypothesis over finite fields
Fouvry, Étienne 1; Kowalski, Emmanuel 2; Michel, Philippe 3; Raju, Chandra Sekhar 4; Rivat, Joël 5; Soundararajan, Kannan 4

1 Laboratoire de Mathématiques d’Orsay Université Paris-Sud CNRS Université Paris-Saclay 91405 Orsay (France)
2 ETH Zürich – D-MATH Rämistrasse 101 8092 Zürich (Switzerland)
3 EPFL Mathgeom-TAN Station 8 C1015 Lausanne (Switzerland)
4 Department of Mathematics 450 Serra Mall, Stanford California 94305 (USA)
5 Institut de Mathématiques de Marseille Case 907 Université d’Aix-Marseille 163, avenue de Luminy 13288 Marseille Cedex 9 (France)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
     author = {Fouvry, \'Etienne and Kowalski, Emmanuel and Michel, Philippe and Raju, Chandra Sekhar and Rivat, Jo\"el and Soundararajan, Kannan},
     title = {On short sums of trace functions},
     journal = {Annales de l'Institut Fourier},
     pages = {423--449},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {67},
     number = {1},
     year = {2017},
     doi = {10.5802/aif.3087},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3087/}
AU  - Fouvry, Étienne
AU  - Kowalski, Emmanuel
AU  - Michel, Philippe
AU  - Raju, Chandra Sekhar
AU  - Rivat, Joël
AU  - Soundararajan, Kannan
TI  - On short sums of trace functions
JO  - Annales de l'Institut Fourier
PY  - 2017
SP  - 423
EP  - 449
VL  - 67
IS  - 1
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3087/
UR  - https://doi.org/10.5802/aif.3087
DO  - 10.5802/aif.3087
LA  - en
ID  - AIF_2017__67_1_423_0
ER  - 
%0 Journal Article
%A Fouvry, Étienne
%A Kowalski, Emmanuel
%A Michel, Philippe
%A Raju, Chandra Sekhar
%A Rivat, Joël
%A Soundararajan, Kannan
%T On short sums of trace functions
%J Annales de l'Institut Fourier
%D 2017
%P 423-449
%V 67
%N 1
%I Association des Annales de l’institut Fourier
%U https://doi.org/10.5802/aif.3087
%R 10.5802/aif.3087
%G en
%F AIF_2017__67_1_423_0
Fouvry, Étienne; Kowalski, Emmanuel; Michel, Philippe; Raju, Chandra Sekhar; Rivat, Joël; Soundararajan, Kannan. On short sums of trace functions. Annales de l'Institut Fourier, Volume 67 (2017) no. 1, pp. 423-449. doi : 10.5802/aif.3087. https://aif.centre-mersenne.org/articles/10.5802/aif.3087/

[1] Birch, Bryan J. How the number of points of an elliptic curve over a fixed prime prime varies, J. Lond. Math. Soc., Volume 43 (1968), pp. 57-60

[2] Blomer, Valentin; Fouvry, Étienne; Kowalski, Emmanuel; Michel, Philippe; Milićević, Djordje On moments of twisted L–functions (http://arxiv.org/abs/1411.4467, to appear in American J. Math.)

[3] Bourgain, Jean A Remark on Solutions of the Pell Equation, Int. Math. Res. Not., Volume 2015 (2015) no. 10, pp. 2841-2855

[4] Bourgain, Jean; Garaev, Moubariz Z. Sumsets of reciprocals in prime fields and multilinear Kloosterman sums, Izv. Math., Volume 78 (2014) no. 4, pp. 656-707 translation from Izv. Ross. Akad. Nauk. Mat. 78 (2014), no. 4, p. 656-707

[5] Burgess, D. A. On character sums and L-series II, Proc. Lond. Math. Soc., Volume 13 (1963), pp. 524-536

[6] Chang, Mei-Chu An estimate of incomplete mixed character sums, An irregular mind (Bolyai Soc. Math. Stud.), Volume 21, Springer, 2010, pp. 243-250

[7] Enflo, Per Some problems in the interface between number theory, harmonic analysis and geometry of Euclidean space, Quaest. Math., Volume 18 (1995) no. 1-3, pp. 309-323

[8] Fouvry, Étienne; Kowalski, Emmanuel; Michel, Philippe Counting sheaves using spherical codes, Math. Res. Lett., Volume 20 (2013) no. 2, pp. 305-323

[9] Fouvry, Étienne; Kowalski, Emmanuel; Michel, Philippe An inverse theorem for Gowers norms of trace functions over F p , Math. Proc. Camb. Philos. Soc., Volume 155 (2013) no. 2, pp. 277-295

[10] Fouvry, Étienne; Kowalski, Emmanuel; Michel, Philippe Algebraic trace functions over the primes, Duke Math. J., Volume 163 (2014) no. 9, pp. 1183-1736 (http://arxiv.org/abs/1211.6043)

[11] Fouvry, Étienne; Kowalski, Emmanuel; Michel, Philippe Algebraic twists of modular forms and Hecke orbits, Geom. Funct. Anal., Volume 25 (2015) no. 2, pp. 580-657 (http://arxiv.org/abs/1207.0617)

[12] Fouvry, Étienne; Kowalski, Emmanuel; Michel, Philippe On the exponent of distribution of the ternary divisor function, Mathematika, Volume 61 (2015) no. 1, pp. 121-144

[13] Fouvry, Étienne; Kowalski, Emmanuel; Michel, Philippe Trace functions over finite fields and their applications, Colloquium de Giorgi 2013 and 2014, Volume 5 (2015), pp. 7-35

[14] Frolenkov, Dmitriĭ A.; Soundararajan, Kannan A generalization of the Pólya-Vinogradov inequality, Ramanujan J., Volume 31 (2013) no. 3, pp. 271-279

[15] Heath-Brown, D. Roger MR review of “Some problems in the interface between number theory, harmonic analysis and geometry of Euclidean space” by P. Enflo MR 1340486 (96h:11079)

[16] Heath-Brown, D. Roger; Pierce, Lillian B. Burgess bounds for short mixed character sums, J. Lond. Math.Soc., Volume 91 (2015) no. 3, pp. 693-708

[17] Iwaniec, Henryk; Kowalski, Emmanuel Analytic Number Theory, American Mathematical Society Colloquium Publications, 53, American Mathematical Society, Providence, RI, 2004, xi+615 pages

[18] Katz, Nicholas M. Gauss sums, Kloosterman sums and monodromy groups, Annals of Mathematics Studies, 116, Princeton University Press, 1988, viii+246 pages

[19] Katz, Nicholas M. Convolution and equidistribution: Sato-Tate theorems of finite-field Mellin transforms, Annals of Mathematics Studies, 180, Princeton University Press, 2012, vi+203 pages

[20] Korobov, N. M. Exponential sums and their applications, Mathematics and its Applications (Soviet Series), 80, Kluwer Academic Publishers Group, 1992, xv+209 pages

[21] Kowalski, Emmanuel; Sawin, W. Kloosterman paths and the shape of exponential sums, Compositio Math., Volume 152 (2016) no. 7, pp. 1489-1516

[22] Livné, Ron The average distribution of cubic exponential sums, J. Reine Angew. Math., Volume 375-376 (1987), pp. 362-379

[23] Vaughan, Robert C. The Hardy–Littlewood method., Cambridge Tracts in Mathematics, 125, Cambridge University Press, 1997, vii+232 pages

[24] Wooley, Trevor D. Vinogradov’s mean value theorem via efficient congruencing, Ann. Math., Volume 175 (2012) no. 3, pp. 1575-1627 | DOI

Cited by Sources: