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PreprintCC BY-NC-ND 4.0arXiv

The optimal van der Corput property for squares

  • Steve Fan
  • Andrew Lott
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Abstract

We prove a sharp quantitative form of the van der Corput property for squares. For all sufficiently large N, we construct a real trigonometric polynomial P(θ)=a0+∑1≤ n≤√Nancos(2πn2θ) satisfying tag1labeleq:abstract−vdc P(θ)≥0 (θ∈ℝ/ℤ), P(0)=1, with 0≤ a0≤exp(−c√(log Nloglog N)), where c>0 is an absolute constant. As a consequence, if A⊆ℕ∩[1,N] has no nonzero square difference, then |A|≤ Nexp(−c√(log Nloglog N)), improving upon a recent result of Green and Sawhney. Our construction builds on previous work of Slijepčević, Slijepčević--Ninčević, and Fan--Lott, and the proof of the upper bound is elementary and self-contained. We also show that every real trigonometric polynomial of the above form satisfying \eqref{eq:abstract-vdc} has a0≥exp(−C√(log Nloglog N)) for an absolute constant C>0. Thus the van der Corput property cannot be used to obtain a power-saving bound for the Furstenberg--Sárközy theorem.

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Cite

BibTeXRIS
BibTeX
@misc{fan2026optimal,
  title = {{The optimal van der Corput property for squares}},
  author = {Steve Fan and Andrew Lott},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.07765},
  eprint = {2610.07765},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.07765},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - The optimal van der Corput property for squares
AU  - Steve Fan
AU  - Andrew Lott
PY  - 2026
DA  - 2026/10/12
PB  - arXiv
DO  - 10.48550/arxiv.2610.07765
UR  - https://arxiv.org/abs/2610.07765
AB  - We prove a sharp quantitative form of the van der Corput property for squares. For all sufficiently large N, we construct a real trigonometric polynomial P(θ)=a_0+∑_1≤ n≤√Na_ncos(2πn^2θ) satisfying tag1labeleq:abstract−vdc P(θ)≥0 (θ∈ℝ/ℤ), P(0)=1, with 0≤ a_0≤exp(−c√(log Nloglog N)), where c>0 is an absolute constant. As a consequence, if A⊆ℕ∩[1,N] has no nonzero square difference, then |A|≤ Nexp(−c√(log Nloglog N)), improving upon a recent result of Green and Sawhney. Our construction builds on previous work of Slijepčević, Slijepčević--Ninčević, and Fan--Lott, and the proof of the upper bound is elementary and self-contained. We also show that every real trigonometric polynomial of the above form satisfying \eqref{eq:abstract-vdc} has a_0≥exp(−C√(log Nloglog N)) for an absolute constant C>0. Thus the van der Corput property cannot be used to obtain a power-saving bound for the Furstenberg--Sárközy theorem.
ER  -