Uniform spacing of zeros of orthogonal polynomials

It is shown that for doubling weights, the zeros of the associated orthogonal polynomials are uniformly spaced in the sense that if $\cos\theta_{m,k}, \theta_{m,k}\in [0, pi]$ are the zeros of the $m$-th orthogonal polynomial associated with $w$, then $\theta_{m,k}-\theta_{m,k+1}\sim 1/m$. It is also shown that for doubling weights, neighboring Cotes numbers are of the same order. Finally, it is proved that these two properties are actually equivalent to the doubling property of the weight function.