PreprintCC BY 4.0arXiv
Improved Weyl bounds on short intervals
Abstract
For an integer d≥ 3, put Δd=min2d−1,d(d−1). Let a/q be reduced, let P(X)=a/qXd+αd−1Xd−1+⋯+α0, and let ℐ be an interval of H≤ q consecutive integers. We prove |∑n∈ℐe(P(n))|≪d,εq1/dHε+H1−1/Δd+ε. Consequently, for every prime p>d, every degree-d polynomial P∈𝔽p[X], and every interval ℐ of H consecutive integers with p1/d<H<p1/(d−1), writing Hd/p=Hu, one has |∑n∈ℐep(P(n))|≪d,εH1−minu/d,1/Δd+ε. This strictly improves throughout the full natural short-interval window the best generic estimate obtained by combining classical Weyl differencing with the optimal Vinogradov mean value theorem.
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BibTeX
@misc{hu2026improved,
title = {{Improved Weyl bounds on short intervals}},
author = {Xiyu Hu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02478},
eprint = {2609.02478},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02478},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Improved Weyl bounds on short intervals
AU - Xiyu Hu
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.02478
UR - https://arxiv.org/abs/2609.02478
AB - For an integer d≥ 3, put Δ_d=min2^d−1,d(d−1). Let a/q be reduced, let P(X)=a/qX^d+α_d−1X^d−1+⋯+α_0, and let ℐ be an interval of H≤ q consecutive integers. We prove |∑_n∈ℐe(P(n))|≪_d,εq^1/dH^ε+H^1−1/Δ_d+ε. Consequently, for every prime p>d, every degree-d polynomial P∈𝔽p[X], and every interval ℐ of H consecutive integers with p^1/d<H<p^1/(d−1), writing H^d/p=H^u, one has |∑n∈ℐe_p(P(n))|≪_d,εH^1−minu/d,1/Δ_d+ε. This strictly improves throughout the full natural short-interval window the best generic estimate obtained by combining classical Weyl differencing with the optimal Vinogradov mean value theorem.
ER -