Turán problem on the union of three intervals of equal length

This paper addresses the Turán extremal problem on the symmetric three-interval set ${\mathrm{\Omega }}_{\lambda }=(-1,1)\cup (\lambda -1,\lambda +1)\cup (-\lambda -1,-\lambda +1)$, $\lambda \ge 0$. We establish a discrete approximation principle relating the continuous Turán problem on ${\mathrm{\Omega }}_{\lambda }$ 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 $\lambda$. We further analyze the dependence of the Turán constant on $\lambda$ and derive upper bounds for all $\lambda >5$.