Asymptotics of Erd\H{o}s's \(L^2\) Lagrange Interpolation Problem: Arcsine Distribution and Airy Endpoint Universality Revisited

Let $x_1,\dots,x_n\in[-1,1]$ be distinct nodes and let
\[
l_k(x)=\prod_{i\neq k}\frac{x-x_i}{x_k-x_i}
\]
denote the associated Lagrange interpolation polynomials. Erd\H{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 $n\to\infty$. 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 place Erd\H{o}s’s problem within the classical framework of minimal-norm interpolation. We interpret $I$ as the squared Hilbert--Schmidt norm of the associated Lagrange interpolation operator and recall that asymptotic minimizers are constrained by the structure and uniqueness theory of minimal $L^2$-norm interpolation schemes. Building on this foundation, we develop a variational approach based on Christoffel functions, orthogonal polynomial asymptotics, and entropy methods to resolve the problem asymptotically.

Our main contributions are as follows:
\begin{enumerate}
\item[(i)] We prove that any asymptotically minimizing sequence of nodes must equidistribute with respect to the arcsine measure on ([-1,1]).

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\item[(ii)] We establish a sharp \(O(1/n)\) lower bound, improving the longstanding \(O((\log n)^2/n)\) estimate of Erd\H{o}s–Szabados–Varma–V\'ertesi.

\item[(iii)] We show that asymptotic minimizers are rigid and structurally constrained, in accordance with classical minimal-norm interpolation theory.

\item[(iv)] 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[(v)] 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 in terms of the Airy kernel.

\item[(vi)] We show that the Legendre–integral nodes are asymptotically optimal and support all theoretical predictions with detailed numerical experiments, including verification of edge rigidity and Airy-type endpoint scaling.