Asymptotics of Erdos’s L2 Lagrange Interpolation Problem: Arcsine Distribution and Airy Endpoint Universality

Let (x_1,dots,x_nin[-1,1]) be distinct nodes and let [ l_k(x)=prod_{ineq k}frac{x-x_i}{x_k-x_i} ] denote the associated Lagrange interpolation polynomials. ErdH{o}s posed the problem of minimizing the functional [ I(x_1,dots,x_n)=int_{-1}^1 sum_{k=1}^n |l_k(x)|^2,dx ] and determining its asymptotic behavior as (ntoinfty). It was known that [ 2-O!left(frac{(log n)^2}{n}right)le inf I le 2-frac{2}{2n-1}, ] with the upper bound attained by nodes related to Legendre polynomials.In this paper, we develop a variational framework based on Christoffel functions, orthogonal polynomial asymptotics, and entropy methods to resolve this problem asymptotically. Our main contributions are:begin{enumerate} item[(i)] We prove that any asymptotically minimizing sequence of nodes must equidistribute with respect to the arcsine measure on ([-1,1]). item[(ii)] We establish a sharp (O(1/n)) lower bound, improving the longstanding (O((log n)^2/n)) result of ErdH{o}s–Szabados–Varma–V’ertesi. item[(iii)] We identify that the leading correction arises from microscopic endpoint regions and formulate an emph{entropy rigidity hypothesis} connecting deterministic minimization to equilibrium log-gas behavior. item[(iv)] Under a conjectured emph{endpoint universality} principle for discrete Christoffel functions, we derive the first-order asymptotic expansion [ inf I = 2 – frac{c}{n} + o!left(frac{1}{n}right), ] with an explicit constant (c>0) expressed via the Airy kernel. item[(v)] We show that the Legendre–integral nodes are asymptotically optimal and rigid, and support all theoretical predictions with detailed numerical experiments, including verification of edge rigidity and Airy-type endpoint scaling. end{enumerate}The expansion in (iv) is conditional on an endpoint universality conjecture (Conjecture~5.1), whose rigorous proof remains an open problem. A complete verification would finalize the asymptotic solution of ErdH{o}s’s interpolation extremal problem and establish a deeper connection to universality in random matrix theory.