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 $Ω_λ= (-1,1) \cup (λ-1,λ+1) \cup (-λ-1,-λ+1)$, $λ\geq 0$. 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 $λ> 5$.