On the Vinogradov mean value. (English, Russian) Zbl 1371.11138

Bourgain, J.

Proc. Steklov Inst. Math. 296, 30-40 (2017); translation in Tr. Mat. Inst. Steklova 296, 36-46 (2017).

Writing \(e(t)=e^{2\pi it}\), for positive integers \(k,s\) and \(x\in{\mathbb R}^k\) let \[ f_k(x,N)=\sum_{1\leq n\leq N} e(nx_1+n^2x_2+\ldots+n^kx_k) \] and \[ J_{s,k}(N)=\int_{[0,1]^k} |f_k(x,N)|^{2s}\,dx_1\cdots dx_k. \] As it is well-known, \(J_{s,k}\) counts the number of integer solutions of the system \(n_1^j+\ldots+n_s^j=n_{s+1}^j+\ldots+n_{2s}^j\) \((1\leq j\leq k)\), where \(1\leq n_i\leq N\) \((i=1,\dots,2s)\). The evaluation of \(J_{s,k}\) is a central problem, with many important applications. The so-called Main Conjecture (going back to Vinogradov) states that \(|J_{s,k}|\ll N^\varepsilon(N^s+N^{2s-k(k+1)/2})\) for all \(\varepsilon>0\). In the paper a discussion is given on a recent work of C. Demeter, L. Guth and the author [Ann. Math. (2) 184, No. 2, 633–682 (2016; Zbl 1408.11083)] where they proved the Main Conjecture using the decoupling theory for curves.

Reviewer: Lajos Hajdu (Debrecen)

MSC:

11P55	Applications of the Hardy-Littlewood method
11L15	Weyl sums
11L07	Estimates on exponential sums
11P05	Waring’s problem and variants

Keywords:

Vinogradov main conjecture; Weyl sums; Riemann zeta function; decoupling theory for curves

Citations:

Zbl 1408.11083

References:

[1]	G. I. Arkhipov, V. N. Chubarikov, and A. A. Karatsuba, Trigonometric Sums in Number Theory and Analysis (W. de Gruyter, Berlin, 2004). · Zbl 1074.11043
[2]	Bennett, J.; Carbery, A.; Tao, T., On the multilinear restriction and Kakeya conjectures, Acta Math., 196, 261-302, (2006) · Zbl 1203.42019
[3]	J. Bourgain, “Decoupling inequalities and some mean-value theorems,” J. Anal. Math. (in press); arXiv: 1406.7862 [math.NT]. · Zbl 1380.42008
[4]	Bourgain, J., Decoupling, exponential sums and the Riemann zeta function, J. Am. Math. Soc., 30, 205-224, (2017) · Zbl 1352.11065
[5]	Bourgain, J.; Demeter, C., The proof of the l2 decoupling conjecture, Ann. Math., Ser. 2, 182, 351-389, (2015) · Zbl 1322.42014
[6]	J. Bourgain and C. Demeter, “Decouplings for curves and hypersurfaces with nonzero Gaussian curvature,” J. Anal. Math. (in press); arXiv: 1409.1634 [math.CA]. · Zbl 1384.42016
[7]	J. Bourgain and C. Demeter, “Mean value estimates for Weyl sums in two dimensions,” arXiv: 1509.05388 [math.CA]. · Zbl 1408.11082
[8]	J. Bourgain, C. Demeter, and L. Guth, “Proof of the main conjecture in Vinogradov’s mean value theorem for degrees higher than three,” arXiv: 1512.01565 [math.NT]. · Zbl 1408.11083
[9]	Bourgain, J.; Guth, L., Bounds on oscillatory integral operators based on multilinear estimates, Geom. Funct. Anal., 21, 1239-1295, (2011) · Zbl 1237.42010
[10]	Ford, K.; Wooley, T. D., On vinogradov’s Mean value theorem: strongly diagonal behaviour via efficient congruencing, Acta Math., 213, 199-236, (2014) · Zbl 1307.11102
[11]	Heath-Brown, D. R., Weyl’s inequality, hua’s inequality, and waring’s problem, J. London Math. Soc., Ser. 2, 38, 216-230, (1988) · Zbl 0619.10046
[12]	Karatsuba, A. A., The Mean value of the modulus of a trigonometric sum, Izv. Akad. Nauk SSSR, Ser. Mat., 37, 1203-1227, (1973)
[13]	Linnik, U. V., On weyl’s sums, Mat. Sb., 12, 28-39, (1943) · Zbl 0063.03578
[14]	Parsell, S. T.; Prendiville, S. M.; Wooley, T. D., Near-optimal Mean value estimates for multidimensional Weyl sums, Geom. Funct. Anal., 23, 1962-2024, (2013) · Zbl 1320.11028
[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, 14-30, (2000) · Zbl 1006.11046
[16]	Stechkin, S. B., On Mean values of the modulus of a trigonometric sum, Tr. Mat. Inst. im. V.A. Steklova, Akad. Nauk SSSR, 134, 283-309, (1975)
[17]	Vaughan, R. C., On waring’s problem for cubes, J. Reine Angew. Math., 365, 122-170, (1986) · Zbl 0574.10046
[18]	R. C. Vaughan, The Hardy-Littlewood Method, 2nd ed. (Cambridge Univ. Press, Cambridge, 1997). · Zbl 0868.11046
[19]	Vinogradov, I. M., New estimates for Weyl sums, Dokl. Akad. Nauk SSSR, 3, 195-198, (1935) · Zbl 0013.05303
[20]	I. M. Vinogradov, The Method of Trigonometrical Sums in the Theory of Numbers (Akad. Nauk SSSR, Moscow, 1947; Interscience Publ., New York, 1954), Tr. Mat. Inst. im. V.A. Steklova, Akad. Nauk SSSR 23. · Zbl 0041.37002
[21]	Wooley, T. D., On vinogradov’s Mean value theorem, Mathematika, 39, 379-399, (1992) · Zbl 0769.11036
[22]	Wooley, T. D., Some remarks on vinogradov’s Mean value theorem and tarry’s problem, Monatsh. Math., 122, 265-273, (1996) · Zbl 0881.11043
[23]	Wooley, T. D., The asymptotic formula in waring’s problem, Int. Math. Res. Not., 2012, 1485-1504, (2012) · Zbl 1267.11104
[24]	Wooley, T. D., Vinogradov’s Mean value theorem via efficient congruencing, Ann. Math., Ser. 2, 175, 1575-1627, (2012) · Zbl 1267.11105
[25]	Wooley, T. D., Vinogradov’s Mean value theorem via efficient congruencing. II, Duke Math. J., 162, 673-730, (2013) · Zbl 1312.11066
[26]	Wooley, T. D., Multigrade efficient congruencing and vinogradov’s Mean value theorem, Proc. London Math. Soc., Ser. 3, 111, 519-560, (2015) · Zbl 1328.11087
[27]	T. D. Wooley, “Approximating the main conjecture in Vinogradov’s mean value theorem,” arXiv: 1401.2932 [math.NT]. · Zbl 1372.11086
[28]	Wooley, T. D., The cubic case of the main conjecture in vinogradov’s Mean value theorem, Adv. Math., 294, 532-561, (2016) · Zbl 1365.11097
[29]	Wooley, T. D., Translation invariance, exponential sums, and waring’s problem, 505-529, (2014) · Zbl 1373.11061