Distances of roots of classical orthogonal polynomials

Let $(P_N{)}_{N\ge 0}$ one of the classical sequences of orthogonal polynomials, i.e., Hermite, Laguerre or Jacobi polynomials. For the roots $z_{1,N},\ldots, z_{N,N}$of$P_N$wederivelowerestimatesfor$\min_{i\ne j}|z_{i,N}-z_{j,N}|$ and the distances from the boundary of the orthogonality intervals. The proofs are based on recent results on the eigenvalues of the covariance matrices in central limit theorems for associated $\beta$-random matrix ensembles where these entities appear as entries, and where the eigenvalues of these matrices are known.