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An improved high-low method for the parabola

  • Zane Kun Li
  • Po-Lam Yung
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Abstract

We demonstrate a modification of the high-low method of Guth-Maldague-Wang that yields a decoupling theorem which gives the sharp estimate for the discrete restriction constant for the parabola and also a diameter-free bound for the three-fold additive energy of rational points on the parabola and circle. These results were recently obtained by Deng-Fan-Guo-Guo-Luo and Cushman-Demeter-Wu, respectively. The novel features in this argument are an orthogonal expansion of F3 and a new square function that also keeps the wavepackets discarded after pruning, resulting in new and more efficient high and low lemmas.

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Cite

BibTeXRIS
BibTeX
@misc{li2026improved,
  title = {{An improved high-low method for the parabola}},
  author = {Zane Kun Li and Po-Lam Yung},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.33949},
  eprint = {2609.33949},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.33949},
}
RIS
TY  - UNPB
TI  - An improved high-low method for the parabola
AU  - Zane Kun Li
AU  - Po-Lam Yung
PY  - 2026
DA  - 2026/09/27
PB  - arXiv
DO  - 10.48550/arxiv.2609.33949
UR  - https://arxiv.org/abs/2609.33949
AB  - We demonstrate a modification of the high-low method of Guth-Maldague-Wang that yields a decoupling theorem which gives the sharp estimate for the discrete restriction constant for the parabola and also a diameter-free bound for the three-fold additive energy of rational points on the parabola and circle. These results were recently obtained by Deng-Fan-Guo-Guo-Luo and Cushman-Demeter-Wu, respectively. The novel features in this argument are an orthogonal expansion of F^3 and a new square function that also keeps the wavepackets discarded after pruning, resulting in new and more efficient high and low lemmas.
ER  -