Reversing a Philosophy: From Counting to Square Functions and Decoupling


Breakthrough work of Bourgain, Demeter, and Guth recently established that decoupling inequalities can prove powerful results on counting integral solutions to systems of Diophantine equations. In this note we demonstrate that in appropriate situations this implication can also be reversed. As a first example, we observe that a count for the number of integral solutions to a system of Diophantine equations implies a discrete decoupling inequality. Second, in our main result we prove an \(L^{2n}\) square function estimate (which implies a corresponding decoupling estimate) for the extension operator associated to a non-degenerate curve in \(\mathbb {R}^n\). The proof is via a combinatorial argument that builds on the idea that if \(\gamma \) is a non-degenerate curve in \(\mathbb {R}^n\), then as long as \(x_1,\ldots , x_{2n}\) are chosen from a sufficiently well-separated set, then \( \gamma (x_1)+\cdots +\gamma (x_n) = \gamma (x_{n+1}) + \cdots + \gamma (x_{2n}) \) essentially only admits solutions in which \(x_1,\ldots ,x_n\) is a permutation of \(x_{n+1},\ldots , x_{2n}\).

This is a preview of subscription content, access via your institution.


  1. 1.

    Arkhipov, G.I., Chubarikov, V.N., Karatsuba, A.A.: Exponent of convergence of the singular integral in the Tarry problem. Dokl. Akad. Nauk SSSR 248(2), 268–272 (1979)

    MathSciNet  Google Scholar 

  2. 2.

    Arkhipov, G.I., Chubarikov, V.N., Karatsuba, A.A.: Trigonometric sums in number theory and analysis. Translated from the 1987 Russian original. De Gruyter Expositions in Mathematics, 39. Berlin (2004)

  3. 3.

    Bak, J.-G., Oberlin, D.M., Seeger, A.: Restriction of Fourier transforms to curves and related oscillatory integrals. Am. J. Math. 131(2), 277–311 (2009)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Bourgain, J., Demeter, C.: A study guide for the \(\ell ^2\) decoupling theorem. Chin. Ann. Math. Ser. B 38(1), 173–200 (2017)

    MathSciNet  Article  Google Scholar 

  5. 5.

    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)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Dendrinos, S., Wright, J.: Fourier restriction to polynomial curves I: a geometric inequality. Am. J. Math. 132(4), 1031–1076 (2010)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Dendrinos, S., Laghi, N., Wright, J.: Universal \(L^p\) improving for averages along polynomial curves in low dimensions. J. Funct. Anal. 257, 1355–1378 (2009)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Drury, S.W.: Restrictions of Fourier transforms to curves. Ann. Inst. Fourier (Grenoble) 35(1), 117–123 (1985)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Fefferman, C.: A note on spherical summation multipliers. Israel J. Math. 15(1), 44–52 (1973)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Gloden, A.: Mehrgradige Gleichungen. P. Nordhoff, Groningen (1944)

    Google Scholar 

  11. 11.

    Hardy, G.H., Wright, E.M.: Introduction to the theory of numbers, 6th edition, revised by D. Oxford University Press, Oxford, R. Heath-Brown and J. H. Silverman (2008)

  12. 12.

    Mordell, L.J.: On a sum analogous to a Gauss’s sum. Quart. J. Math. 3(1), 161–167 (1932)

    Article  Google Scholar 

  13. 13.

    Pierce, L. B.: The Vinogradov Mean Value Theorem [after Wooley, and Bourgain, Demeter, Guth]. Séminaire Bourbaki (volume 69, 2016/2017, exposé 1134), Astérisque, (2019) volume 407

  14. 14.

    Stein, E.M., Weiss, G.: Introduction to Fourier Analysis on Euclidean Spaces Princeton Mathematical Series, No 32. Princeton University Press, Princeton, NJ (1971)

    Google Scholar 

  15. 15.

    Vaughan, R.C., Wooley, T.D.: A special case of Vinogradov’s mean value theorem. Acta Arithmetica LXXIX. 3, 193–204 (1997)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Wooley, T.D.: The cubic case of the main conjecture in Vinogradov’s mean value theorem. Adv. Math. 294, 532–561 (2016)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Wooley, T.D.: Nested efficient congruencing and relatives of Vinogradov’s mean value theorem. Proc. London Math. Soc. 118(4), 942–1016 (2019)

Download references


We thank the American Institute of Mathematics for funding our collaboration in the context of a SQuaRE workshop series. Gressman has been partially supported by NSF Grant DMS-1764143. Pierce has been partially supported by NSF CAREER Grant DMS-1652173, a Sloan Research Fellowship, and as a von Neumann Fellow at the Institute for Advanced Study, by the Charles Simonyi Endowment and NSF Grant No. 1128155. Yung was partially supported by a General Research Fund CUHK14303817 from the Hong Kong Research Grant Council, and a direct grant for research from the Chinese University of Hong Kong (Grant No. 4053341).

Author information



Corresponding author

Correspondence to Philip T. Gressman.

Additional information

Dedicated to Elias M. Stein, in deep appreciation of his generous teaching and clear-sighted vision in harmonic analysis.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Gressman, P.T., Guo, S., Pierce, L.B. et al. Reversing a Philosophy: From Counting to Square Functions and Decoupling. J Geom Anal (2021).

Download citation


  • Decoupling inequalities
  • Diophantine equations
  • Square functions

Mathematics Subject Classification

  • 42B20
  • 42B25
  • 11D45