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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 \(x_{n+1},\ldots , x_{2n}\).

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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).

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Correspondence to Philip T. Gressman.

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

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Gressman, P.T., Guo, S., Pierce, L.B. et al. Reversing a Philosophy: From Counting to Square Functions and Decoupling. J Geom Anal 31, 7075–7095 (2021).

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  • Decoupling inequalities
  • Diophantine equations
  • Square functions

Mathematics Subject Classification

  • 42B20
  • 42B25
  • 11D45