等长三个区间之并集上的Turán问题
Turán problem on the union of three intervals of equal length
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中文总结 AI 辅助
本文针对对称三区间集上的Turán极值问题，建立离散逼近原理将其转化为等价的有限半定规划问题，计算了整数和有理参数λ对应的Turán常数，分析了其对λ的依赖性并推导了λ>5时的上界。
中文摘要 AI 辅助
本文研究对称三区间集Ω_λ=(-1,1)∪(λ-1,λ+1)∪(-λ-1,-λ+1)（λ≥0）上的Turán极值问题。我们建立了一个离散逼近原理，将Ω_λ上的连续Turán问题与相关有限离散集上的离散Turán问题关联起来。这使得连续问题可精确表示为有限非负三角多项式问题的极限，而这些问题等价于有限半定规划。利用该框架，我们计算了整数和有理参数λ对应的Turán常数，进一步分析了Turán常数对λ的依赖性，并推导了所有λ>5时的上界。
英文摘要
This paper addresses the Turán extremal problem on the symmetric three-interval set $Ω_λ= (-1,1) \cup (λ-1,λ+1) \cup (-λ-1,-λ+1)$, $λ\geq 0$. 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 $λ> 5$.
