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Source. Proposition 3.1, stated on p. 3 and proved on p. 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 (p. 6) was read for its structure. Nothing here is independently reviewed.

Setting

Section 2, p. 2. For $\tau>0$ the Bernstein space $B_\tau$ is the space of entire functions $f$ of exponential type at most $\tau$ with $\lVert f\rVert_{\infty,\mathbb R}=\sup_{x\in\mathbb R}\lvert f(x)\rvert<\infty$; type at most $\tau$ means that for every $\delta>0$ some $A_\delta<\infty$ gives $\lvert f(z)\rvert\le A_\delta e^{(\tau+\delta)\lvert z\rvert}$.

Statement

Proposition 3.1 (p. 3). For every $C>0$ there are $\eta>0$ and an integer $L\ge1$ such that, for every multiset $u_1,\ldots,u_L\in\mathbb R$ with

$$\operatorname{diam}\{u_1,\ldots,u_L\}\le\pi(1+\eta)L,$$

there are real labels $a_1,\ldots,a_L\in[-1,1]$ such that no complex-valued $f\in B_1$ with $\lVert f\rVert_{\infty,\mathbb R}\le C$ satisfies $f(u_j)=a_j$ for all $j=1,\ldots,L$.

The paper calls it the main compactness consequence of Theorem 2.2 and the only place where the strict density bound is used (p. 3). Section 6.1 (p. 9) notes that no quantitative control of interpolation constants near the critical density is needed, and Section 6.3 that $L(C)$ and $\eta(C)$ come from compactness with no explicit values.

Proof pointer

P. 6. For $0<C<1$ take $L=1$ and the label $a_1=1$. For $C\ge1$, if the proposition failed, a multiset that interpolates every $[-1,1]$ datum could have no repeated point (labels $1$ and $-1$ on two copies), so along $\eta_k\downarrow0$, $L_k\to\infty$ one gets sets translated to have convex hull $[0,T_k]$ with $T_k\le\pi(1+\eta_k)L_k$, and Lemma 3.2 yields a separated interpolation sequence for $B_1$ of upper uniform density at least $1/\pi$, against Theorem 2.2.

Dependencies

Lemma 3.2 and Theorem 2.2 (p. 3), Beurling's theorem on the real line: for $\tau>0$ a separated $\Lambda\subset\mathbb R$ is an interpolation sequence for $B_\tau$ if and only if its upper uniform density $D^+(\Lambda)$ is below $\tau/\pi$; the paper cites it from Beurling's collected works and Ortega-Cerdà and Seip, J. Funct. Anal. 162 (1999), Theorem 1.

Bears on

• #1133: the local obstruction that the paper plants on blocks of nodes to prove Theorem 1.1; on its own it concerns functions of exponential type, not polynomials.