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Turán problem on the union of three intervals of equal length
Abstract
This paper addresses the Turán extremal problem on the symmetric three-interval set Ωλ= (−1,1) ∪ (λ−1,λ+1) ∪ (−λ−1,−λ+1), λ≥ 0. We establish a discrete approximation principle relating the continuous Turán problem on Ωλ to the discrete Turán problem on an associated finite discrete set. This enables the continuous problem to be exactly expressed as a limit of finite nonnegative trigonometric polynomial problems, which are equivalent to finite semidefinite programs. Using this framework, we compute the Turán constant for integer and rational parameters λ. We further analyze the dependence of the Turán constant on λ and derive upper bounds for all λ> 5.
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BibTeX
@misc{fu2026turán,
title = {{Turán problem on the union of three intervals of equal length}},
author = {Xiao-Ye Fu and Yue-Kun Li and Wei-Jie Wang and Xuan Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.15643},
eprint = {2608.15643},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.15643},
}
RIS
TY - UNPB
TI - Turán problem on the union of three intervals of equal length
AU - Xiao-Ye Fu
AU - Yue-Kun Li
AU - Wei-Jie Wang
AU - Xuan Wang
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2608.15643
UR - https://arxiv.org/abs/2608.15643
AB - This paper addresses the Turán extremal problem on the symmetric three-interval set Ω_λ= (−1,1) ∪ (λ−1,λ+1) ∪ (−λ−1,−λ+1), λ≥ 0. We establish a discrete approximation principle relating the continuous Turán problem on Ω_λ to the discrete Turán problem on an associated finite discrete set. This enables the continuous problem to be exactly expressed as a limit of finite nonnegative trigonometric polynomial problems, which are equivalent to finite semidefinite programs. Using this framework, we compute the Turán constant for integer and rational parameters λ. We further analyze the dependence of the Turán constant on λ and derive upper bounds for all λ> 5.
ER -