Improved L^p maximal estimates for higher-degree Weyl sums

Abstract

For each integer k≥3, we establish the following estimate for all 1≤ p<∞ and ε>0: |sup_t∈𝕋|∑_n=1^N e(nx+n^kt)||_L^p(𝕋) ≪_k,p,ε N^{max{1−1/K_k, 1−1/p}+ε}, where K_k=min{2^k−1,k(k+1)/2}, e(z)=e^2πiz, and 𝕋=ℝ/ℤ. For k≥5, this improves our earlier bound \cite{chen-miao-yuan-zhao} for 1≤ p<D_k, where D_k=min{2^k−1,k(k−1)}, and, together with the corresponding lower bound, extends the sharp range up to N^ε from p≥ D_k to p≥ K_k. The main new ingredient is a mixed minor-arc moment estimate obtained before taking the time supremum. The minor-arc maximal estimate then follows from a smooth reproducing kernel argument, while the major arcs are handled directly by Poisson summation and classical complete exponential-sum estimates. The reduction from suprema to mean values also yields a short and direct proof of improved L^p estimates for maximal operators on Weyl sums with general polynomial phases, strengthening \cite[Theorem 1.1]{baker-chen-shparlinski}.

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BibTeX RIS

BibTeX

@misc{chen2026improved,
  title = {{Improved $L^p$ maximal estimates for higher-degree Weyl sums}},
  author = {Xuezhi Chen and Changxing Miao and Jiye Yuan and Tengfei Zhao},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.23379},
  eprint = {2610.23379},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.23379},
}

RIS

TY  - UNPB
TI  - Improved L^p maximal estimates for higher-degree Weyl sums
AU  - Xuezhi Chen
AU  - Changxing Miao
AU  - Jiye Yuan
AU  - Tengfei Zhao
PY  - 2026
DA  - 2026/10/16
PB  - arXiv
DO  - 10.48550/arxiv.2610.23379
UR  - https://arxiv.org/abs/2610.23379
AB  - For each integer k≥3, we establish the following estimate for all 1≤ p<∞ and ε>0: |sup_t∈𝕋|∑_n=1^N e(nx+n^kt)||_L^p(𝕋) ≪_k,p,ε N^max{1−1/K_k, 1−1/p}+ε, where K_k=min{2^k−1,k(k+1)/2}, e(z)=e^2πiz, and 𝕋=ℝ/ℤ. For k≥5, this improves our earlier bound \cite{chen-miao-yuan-zhao} for 1≤ p<D_k, where D_k=min{2^k−1,k(k−1)}, and, together with the corresponding lower bound, extends the sharp range up to N^ε from p≥ D_k to p≥ K_k. The main new ingredient is a mixed minor-arc moment estimate obtained before taking the time supremum. The minor-arc maximal estimate then follows from a smooth reproducing kernel argument, while the major arcs are handled directly by Poisson summation and classical complete exponential-sum estimates. The reduction from suprema to mean values also yields a short and direct proof of improved L^p estimates for maximal operators on Weyl sums with general polynomial phases, strengthening \cite[Theorem 1.1]{baker-chen-shparlinski}.
ER  -

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