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Decoupling for fractal subsets of the parabola

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Abstract

We consider decoupling for a fractal subset of the parabola. We reduce studying \(l^{2}L^{p}\) decoupling for a fractal subset on the parabola \(\{(t, t^2) : 0 \le t \le 1\}\) to studying \(l^{2}L^{p/3}\) decoupling for the projection of this subset to the interval [0, 1]. This generalizes the decoupling theorem of Bourgain-Demeter in the case of the parabola. Due to the sparsity and fractal like structure, this allows us to improve upon Bourgain–Demeter’s decoupling theorem for the parabola. In the case when p/3 is an even integer we derive theoretical and computational tools to explicitly compute the associated decoupling constant for this projection to [0, 1]. Our ideas are inspired by the recent work on ellipsephic sets by Biggs (arXiv:1912.04351, 2019 and Acta Arith. 200(4):331–348, 2021) using nested efficient congruencing.

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Acknowledgements

JD was partially supported by “La Caixa” Fellowship LCF/ BQ/ AA17/ 11610013. RG was partially supported by the Eric and Wendy Schmidt Postdoctoral Award. AJ was supported by DFG-research fellowship JA 2512/3-1. ZL is supported by NSF grant DMS-1902763. ZL is also grateful to the Department of Mathematics at the University of Chicago and the University of California, Los Angeles for their hospitality when he visited in February 2020. The authors would also like to thank Iqra Altaf, Kirsti Biggs, Julia Brandes, Ciprian Demeter, Bingyang Hu, and Terence Tao for helpful comments, discussions, and suggestions.

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Authors and Affiliations

  1. Department of Mathematics, Princeton University, Princeton, NJ, 08544, USA

    Alan Chang

  2. Department of Mathematics, University of California, Los Angeles, 520 Portola Plaza, Los Angeles, CA, 90095, USA

    Jaume de Dios Pont, Rachel Greenfeld & José Madrid

  3. Department of Mathematics, Koç University, Rumelifeneri Yolu, Sariyer, 34450, Istanbul, Turkey

    Asgar Jamneshan

  4. Department of Mathematics, Indiana University Bloomington, Bloomington, IN, 47405, USA

    Zane Kun Li

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  1. Alan Chang
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  2. Jaume de Dios Pont
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Correspondence to Zane Kun Li.

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Chang, A., Dios Pont, J.d., Greenfeld, R. et al. Decoupling for fractal subsets of the parabola. Math. Z. 301, 1851–1879 (2022). https://doi.org/10.1007/s00209-021-02950-0

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