Geodesic
On the Vinogradov mean value
Informatics and Automation, Analytic and combinatorial number theory, Tome 296 (2017), pp. 36-46 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We discuss the recent work of C. Demeter, L. Guth and the author on the proof of the Vinogradov Main Conjecture using the decoupling theory for curves. 
@article{TRSPY_2017_296_a2,
     author = {J. Bourgain},
     title = {On the {Vinogradov} mean value},
     journal = {Informatics and Automation},
     pages = {36--46},
     year = {2017},
     volume = {296},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2017_296_a2/}
}
TY  - JOUR
AU  - J. Bourgain
TI  - On the Vinogradov mean value
JO  - Informatics and Automation
PY  - 2017
SP  - 36
EP  - 46
VL  - 296
UR  - http://geodesic.mathdoc.fr/item/TRSPY_2017_296_a2/
LA  - ru
ID  - TRSPY_2017_296_a2
ER  - 
%0 Journal Article
%A J. Bourgain
%T On the Vinogradov mean value
%J Informatics and Automation
%D 2017
%P 36-46
%V 296
%U http://geodesic.mathdoc.fr/item/TRSPY_2017_296_a2/
%G ru
%F TRSPY_2017_296_a2
J. Bourgain. On the Vinogradov mean value. Informatics and Automation, Analytic and combinatorial number theory, Tome 296 (2017), pp. 36-46. http://geodesic.mathdoc.fr/item/TRSPY_2017_296_a2/

[1] Arkhipov G. I., Chubarikov V. N., Karatsuba A. A., Trigonometric sums in number theory and analysis, W. de Gruyter, Berlin, 2004 | MR | Zbl

[2] Bennett J., Carbery A., Tao T., “On the multilinear restriction and Kakeya conjectures”, Acta math., 196:2 (2006), 261–302 | DOI | MR | Zbl

[3] Bourgain J., “Decoupling inequalities and some mean-value theorems”, J. anal. math. (to appear)

[4] Bourgain J., “Decoupling, exponential sums and the Riemann zeta function”, J. Amer. Math. Soc., 30:1 (2017), 205–224 ; arXiv: 1408.5794[math.NT] | DOI | MR | Zbl

[5] Bourgain J., Demeter C., “The proof of the $l^2$ Decoupling Conjecture”, Ann. Math. Ser. 2, 182:1 (2015), 351–389 | DOI | MR | Zbl

[6] Bourgain J., Demeter C., “Decouplings for curves and hypersurfaces with nonzero Gaussian curvature”, J. anal. math. (to appear)

[7] Bourgain J., Demeter C., Mean value estimates for Weyl sums in two dimensions, E-print, 2015, arXiv: 1509.05388[math.CA]

[8] Bourgain J., Demeter C., Guth L., Proof of the main conjecture in Vinogradov's mean value theorem for degrees higher than three, E-print, 2015, arXiv: 1512.01565[math.NT] | MR

[9] Bourgain J., Guth L., “Bounds on oscillatory integral operators based on multilinear estimates”, Geom. Funct. Anal., 21:6 (2011), 1239–1295 | DOI | MR | Zbl

[10] Ford K., Wooley T. D., “On Vinogradov's mean value theorem: strongly diagonal behaviour via efficient congruencing”, Acta math., 213:2 (2014), 199–236 | DOI | MR | Zbl

[11] Heath-Brown D. R., “Weyl's inequality, Hua's inequality, and Waring's problem”, J. London Math. Soc. Ser. 2, 38:2 (1988), 216–230 | DOI | MR | Zbl

[12] Karatsuba A. A., “Srednee znachenie modulya trigonometricheskoi summy”, Izv. AN SSSR. Ser. mat., 37:6 (1973), 1203–1227 | MR | Zbl

[13] Linnik U. V., “On Weyl's sums”, Mat. sb., 12(54):1 (1943), 28–39 | MR | Zbl

[14] Parsell S. T., Prendiville S. M., Wooley T. D., “Near-optimal mean value estimates for multidimensional Weyl sums”, Geom. Funct. Anal., 23:6 (2013), 1962–2024 | DOI | MR | Zbl

[15] Robert O., Sargos P., “Un théorème de moyenne pour les sommes d'exponentielles. Application à l'inégalité de Weil”, Publ. Inst. Math. Nouv. sér., 67 (2000), 14–30 | MR | Zbl

[16] Stechkin S. B., “O srednikh znacheniyakh modulya trigonometricheskoi summy”, Tr. MIAN, 134, 1975, 283–309 | MR | Zbl

[17] Vaughan R. C., “On Waring's problem for cubes”, J. Reine Angew. Math., 365 (1986), 122–170 | MR | Zbl

[18] Vaughan R. C., The Hardy–Littlewood method, 2nd ed., Cambridge Univ. Press, Cambridge, 1997 | MR | Zbl

[19] Vinogradov I. M., “Novye otsenki summ Veilya”, DAN SSSR, 3:5 (1935), 195–198 | Zbl

[20] Vinogradov I. M., Metod trigonometricheskikh summ v teorii chisel, Tr. MIAN, 23, Izd-vo AN SSSR, M.–L., 1947 | MR | Zbl

[21] Wooley T. D., “On Vinogradov's mean value theorem”, Mathematika, 39:2 (1992), 379–399 | DOI | MR | Zbl

[22] Wooley T. D., “Some remarks on Vinogradov's mean value theorem and Tarry's problem”, Monatsh. Math., 122:3 (1996), 265–273 | DOI | MR | Zbl

[23] Wooley T. D., “The asymptotic formula in Waring's problem”, Int. Math. Res. Not., 2012:7 (2012), 1485–1504 | DOI | MR | Zbl

[24] Wooley T. D., “Vinogradov's mean value theorem via efficient congruencing”, Ann. Math. Ser. 2, 175:3 (2012), 1575–1627 | DOI | MR | Zbl

[25] Wooley T. D., “Vinogradov's mean value theorem via efficient congruencing. II”, Duke Math. J., 162:4 (2013), 673–730 | DOI | MR | Zbl

[26] Wooley T. D., “Multigrade efficient congruencing and Vinogradov's mean value theorem”, Proc. London Math. Soc. Ser. 3, 111:3 (2015), 519–560 ; arXiv: 1310.8447[math.NT] | DOI | MR | Zbl

[27] Wooley T. D., Approximating the main conjecture in Vinogradov's mean value theorem, E-print, 2014, arXiv: 1401.2932[math.NT]

[28] Wooley T. D., “The cubic case of the main conjecture in Vinogradov's mean value theorem”, Adv. Math., 294 (2016), 532–561 ; arXiv: 1401.3150[math.NT] | DOI | MR | Zbl

[29] Wooley T. D., “Translation invariance, exponential sums, and Waring's problem”, Proc. Int. Congr. Math. (Seoul, Korea, 2014), KM Kyung Moon SA, Seoul, 2014, 505–529 ; arXiv: 1404.3508v1[math.NT] | MR