On integer solutions of Parsell–Vinogradov systems

  • Shaoming Guo
  • Ruixiang ZhangEmail author


We prove a sharp upper bound on the number of integer solutions of the Parsell–Vinogradov system in every dimension \(d\ge 2\).

Mathematics Subject Classification

Primary 11L07 Secondary 42A45 



The authors thank Ciprian Demeter for numerous discussions on related topics. The first author thanks Julia Brandes and Lillian Pierce for discussions on applications of their result. Part of this work is contained in the PhD thesis [43] of the second author. He would like to thank his advisor Peter Sarnak for a lot of very helpful discussions. Part of this material is based upon work supported by the National Science Foundation under Grant No. DMS-1440140 while the first author was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring semester of 2017. The work of the second author is supported by the National Science Foundation under Grant No. 1638352 and the James D. Wolfensohn Fund.


  1. 1.
    Arhipov, G.I., Karacuba, A.A., Cubarikov, V.N.: Multiple trigonometric sums (Russian). Trudy Mat. Inst. Steklov. 151, 128 (1980)Google Scholar
  2. 2.
    Bennett, J., Bez, N., Flock, T., Lee, S.: Stability of Brascamp-Lieb constant and applications. Amer. J. Math. 140(2), 543–569 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bennett, J., Carbery, A., Christ, M., Tao, T.: Finite bounds for Hölder–Brascamp–Lieb multilinear inequalities. Math. Res. Lett. 17(4), 647–666 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bennett, J., Carbery, T., Tao, T.: On the multilinear restriction and Kakeya conjectures. Acta Math. 196(2), 261–302 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bourgain, J.: Moment inequalities for trigonometric polynomials with spectrum in curved hypersurfaces. Isr. J. Math. 193(1), 441–458 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bourgain, J., Demeter, C.: The proof of the \(l^2\) decoupling conjecture. Ann. Math. 182(1), 351–389 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bourgain, J., Demeter, C.: Decouplings for surfaces in \(\mathbb{R}^4\). J. Funct. Anal. 270(4), 1299–1318 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bourgain, J., Demeter, C.: Decouplings for curves and hypersurfaces with nonzero Gaussian curvature. J. Anal. Math. 133, 279–311 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bourgain, J., Demeter, C.: Mean value estimates for Weyl sums in two dimensions. J. Lond. Math. Soc. (2) 94(3), 814–838 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bourgain, J., Demeter, C., Guo, S.: Sharp bounds for the cubic Parsell–Vinogradov system in two dimensions. Adv. Math. 320, 827–875 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Bourgain, J., Demeter, C., Guth, L.: Proof of the main conjecture in Vinogradov’s mean value theorem for degrees higher than three. Ann. Math. (2) 184(2), 633–682 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Bourgain, J., Guth, L.: Bounds on oscillatory integral operators based on multilinear estimates. Geom. Funct. Anal. 21(6), 1239–1295 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Brandes, J., Wooley, T.: Vinogradov systems with a slice off. Mathematika 63(3), 797–817 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Demeter, C., Guo, S., Shi, F.: Sharp decouplings for three dimensional manifolds in \(R^5\). Rev. Mat. Iberoamericana (to appear). arXiv:1609.04107
  15. 15.
    Ford, K.: Vinogradov’s integral and bounds for the Riemann zeta function. Proc. Lond. Math. Soc. (3) 85(3), 565–633 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Ford, K., Wooley, T.: On Vinogradov’s mean value theorem: strongly diagonal behaviour via efficient congruencing. Acta Math. 213, 199–236 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Garrigós, G., Seeger, A.: On plate decompositions of cone multipliers. Proc. Edinb. Math. Soc. (2) 52(3), 631–651 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Garrigós, G., Seeger, A.: A mixed norm variant of Wolff’s inequality for paraboloids. Harmonic analysis and partial differential equations, pp. 179–197, Contemp. Math., vol. 505, Amer. Math. Soc., Providence, RI (2010)Google Scholar
  19. 19.
    Guo, S., Zorin-Kranich, P.: Decoupling for moment manifolds associated to Arkhipov–Chubarikov–Karatsuba systems (2018). arXiv:1811.02207
  20. 20.
    Guth, L.: A short proof of the multilinear Kakeya inequality. Math. Proc. Camb. Philos. Soc. 158(1), 147–153 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Heath-Brown, R., Pierce, L.: Burgess bounds for short mixed character sums. J. Lond. Math. Soc. 91(3), 693–708 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Łaba, I., Pramanik, M.: Wolff’s inequality for hypersurfaces. Collect. Math. Extra, 293–326 (2006)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Łaba, I., Wolff, T.: A local smoothing estimate in higher dimensions. J. Anal. Math. 88, 149–171 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Li, Z.: Effective \(l^2\) decoupling for the parabola. arXiv:1711.01202
  25. 25.
    Milnor, J.: On the Betti numbers of real varieties. Proc. Am. Math. Soc. 15, 275–280 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Oh, C.: Decouplings for \(d\)-dimensional surfaces in \(R^{2d}\). arXiv:1609.02022
  27. 27.
    Oleinik, O.A., Petrovskii, I.G.: On the topology of real algebraic surfaces. Izvestiya Akad. Nauk SSSR. Ser. Mat. 13, 389–402 (1949)MathSciNetGoogle Scholar
  28. 28.
    Parsell, S.T.: The density of rational lines on cubic hypersurfaces. Trans. Am. Math. Soc. 352(11), 5045–5062 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Parsell, S.T.: A generalization of Vinogradov’s mean value theorem. Proc. Lond. Math. Soc. (3) 91(1), 1–32 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Parsell, S.T., Prendiville, S.M., Wooley, T.D.: Near-optimal mean value estimates for multidimensional Weyl sums. Geom. Funct. Anal. 23(6), 1962–2024 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Pierce, L.: Burgess bounds for multi-dimensional short mixed character sums. J. Number Theory 163, 172–210 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Pierce, L.: The Vinogradov Mean Value Theorem [after Wooley, and Bourgain, Demeter and Guth]. arXiv:1707.00119
  33. 33.
    Thom, R.: Sur l’homologie des variétés algbriques réelles. Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse). Princeton University Press, Princeton, NJ, pp. 255–265 (1965)Google Scholar
  34. 34.
    Wongkew, R.: Volumes of tubular neighbourhoods of real algebraic varieties. Pac. J. Math. 159(1), 177–184 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Wolff, T.: Local smoothing type estimates on \(L^p\) for large \(p\). Geom. Funct. Anal. 10(5), 1237–1288 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Wooley, T.: Large improvements in Waring’s problem. Ann. Math. (2) 135(1), 131–164 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Wooley, T.: Vinogradov’s mean value theorem via efficient congruencing. Ann. Math. 175, 1575–1627 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Wooley, T.: Vinogradov’s mean value theorem via efficient congruencing. II. Duke Math. J. 162, 673–730 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Wooley, T.: Mean value estimates for odd cubic Weyl sums. Bull. Lond. Math. Soc. 47(6), 946–957 (2015)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Wooley, T.: The cubic case of the main conjecture in Vinogradov’s mean value theorem. Adv. Math. 294, 532–561 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Wooley, T.: Approximating the main conjecture in Vinogradov’s mean value theorem. Mathematika 63(1), 292–350 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Wooley, T.: Nested efficient congruencing and relatives of Vinogradov’s mean value theorem. arxiv:1708.01220
  43. 43.
    Zhang, R.: Perturbed Brascamp–Lieb inequalities and application to Parsell–Vinogradov systems. Academic dissertations (Ph.D.). Princeton University, Princeton, NJGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of MathematicsInstitute for Advanced StudyPrincetonUSA
  2. 2.Department of MathematicsIndiana UniversityBloomingtonUSA
  3. 3.Department of MathematicsUniversity of Wisconsin MadisonMadisonUSA

Personalised recommendations