---

Source. Lemma 3.2, stated on p. 4 and proved on pp. 4-6, of A Bernstein-density proof of Erdős's robust interpolation obstruction, draft manuscript dated 29 April 2026, no author byline, https://www.ulam.ai/research/erdos1133.pdf, as recorded on the source card. An unrefereed manuscript.

Read depth. Claims checked: the statement and the definitions it uses were read clause by clause on the print; the proof (pp. 4-6) was read for its structure, not checked step by step. Nothing here is independently reviewed.

Setting

Section 2, pp. 2-3. $B_1$ is the Bernstein space of entire functions of exponential type at most $1$ bounded on $\mathbb R$, with $\lVert f\rVert_{\infty,\mathbb R}=\sup_{\mathbb R}\lvert f\rvert$. A real sequence $\Lambda$ is separated if $\inf\{\lvert\lambda-\mu\rvert:\lambda\ne\mu\}>0$; its upper uniform Beurling density is

$$D^+(\Lambda)=\limsup_{R\to\infty}\sup_{a\in\mathbb R} \frac{\#(\Lambda\cap[a,a+R])}{R}.$$

Definition 2.1 (p. 3): a separated real $\Lambda$ is an interpolation sequence for $B_\tau$ if every bounded complex sequence $(a_\lambda)$ is $(f(\lambda))$ for some $f\in B_\tau$.

Statement

Lemma 3.2 (p. 4). Let $C\ge1$. Suppose $L_k\to\infty$, $\eta_k\downarrow0$, and finite sets

$$U_k\subset[0,T_k],\qquad \#U_k=L_k,\qquad T_k\le\pi(1+\eta_k)L_k,$$

have the property that every assignment $a:U_k\to[-1,1]$ is realized by some $f\in B_1$ with $\lVert f\rVert_{\infty,\mathbb R}\le C$ and $f(u)=a(u)$ for $u\in U_k$. Then some separated infinite sequence $\Lambda\subset\mathbb R$ is an interpolation sequence for $B_1$ with $D^+(\Lambda)\ge1/\pi$.

Proof pointer

Pp. 4-6. Bernstein's inequality makes the $U_k$ uniformly $2/C$-separated. Averaging the translates of $U_k$ over $[0,T_k]$ gives measures on the compact space of $2/C$-separated closed sets whose weak limit $\nu$ is translation invariant with intensity at least $1/\pi$. Montel's theorem transfers the bounded interpolation property to every configuration in the support of $\nu$, first on finite subsets and then by a diagonal limit, with constant at most $2C$ for complex data; an ergodic component of intensity at least $1/\pi$ and Birkhoff's theorem give a typical configuration of density at least $1/\pi$.

Dependencies

The Bernstein inequality and the strip estimate for $B_\tau$ (p. 2), Montel's theorem, the ergodic decomposition and the continuous-parameter Birkhoff theorem.

Bears on

• #1133: the compactness step behind Proposition 3.1, and through it Theorem 1.1; it says nothing about polynomials on its own.