PreprintCC BY 4.0arXiv
Sets whose differences avoid a bracket quadratic
Abstract
Suppose a set of integers A⊆{1,…,N} has no solutions to a−a'=n⌊√[3]2n⌋, for distinct a,a'∈ A, and n∈ ℕ. We show that |A|≪ N1−c for some absolute constant c>0. To do this, we prove quantitative bounds on the van der Corput property for certain sets of bracket quadratics. This comes as a consequence of establishing exponential sum estimates for these sets, utilising a theorem of Green and Tao on the quantitative equidistribution of polynomial orbits on nilmanifolds, closely following the approach of Neale who went on to prove a Waring-type result. We also extend our result to differences avoiding a family of bracket polynomials (also known as generalised polynomials).
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BibTeX
@misc{younis2026sets,
title = {{Sets whose differences avoid a bracket quadratic}},
author = {Khalid Younis},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.30078},
eprint = {2608.30078},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30078},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Sets whose differences avoid a bracket quadratic
AU - Khalid Younis
PY - 2026
DA - 2026/08/30
PB - arXiv
DO - 10.48550/arxiv.2608.30078
UR - https://arxiv.org/abs/2608.30078
AB - Suppose a set of integers A⊆{1,…,N} has no solutions to a−a'=n⌊√[3]2n⌋, for distinct a,a'∈ A, and n∈ ℕ. We show that |A|≪ N^1−c for some absolute constant c>0. To do this, we prove quantitative bounds on the van der Corput property for certain sets of bracket quadratics. This comes as a consequence of establishing exponential sum estimates for these sets, utilising a theorem of Green and Tao on the quantitative equidistribution of polynomial orbits on nilmanifolds, closely following the approach of Neale who went on to prove a Waring-type result. We also extend our result to differences avoiding a family of bracket polynomials (also known as generalised polynomials).
ER -