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Source. Lemma 4.1, stated on p. 7 and proved on pp. 7-8, 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, its setting and the proof (pp. 7-8) were read on the print. Nothing here is independently reviewed.

Setting

Section 4, p. 7. Fix $C>0$, take $\eta>0$ and $L\in\mathbb N$ from Proposition 3.1, and choose

$$0<\varepsilon<\min\Bigl\{\frac12,\ \frac{\eta}{2\bigl(1+(1+\eta)L\bigr)}\Bigr\}, \tag{6}$$

which gives $(\eta-\varepsilon)/\bigl((1+\eta)L\bigr)>\varepsilon$ (7). Put $D_n=\lceil(1+\varepsilon)n\rceil$. For nodes $x_1,\ldots,x_n\in[-1,1]$ let $\theta_i=\arccos x_i\in[0,\pi]$, indexed so that $0\le\theta_1\le\cdots\le\theta_n\le\pi$. The full blocks are $B_b=\{(b-1)L+1,\ldots,bL\}$ for $b=1,\ldots,N$, $N=\lfloor n/L\rfloor$, the fewer than $L$ leftover indices being ignored; $B_b$ has angular span $h_b=\theta_{bL}-\theta_{(b-1)L+1}$ and is good when

$$D_nh_b\le\pi(1+\eta)L. \tag{8}$$

Statement

Lemma 4.1 (p. 7). For all sufficiently large $n$, the number $G$ of good full blocks satisfies $G>\varepsilon n$.

Proof pointer

Pp. 7-8. The block spans sum to at most $\pi$, so fewer than $D_n/\bigl((1+\eta)L\bigr)$ blocks are bad; with $D_n\le(1+\varepsilon)n+1$ this leaves $G\ge n(\eta-\varepsilon)/\bigl((1+\eta)L\bigr)-O(1)$, and (7) makes the coefficient of $n$ exceed $\varepsilon$.

Dependencies

Only the choice (6) and the definitions above; the constants $\eta$ and $L$ come from Proposition 3.1.

Bears on

• #1133: the counting step in the proof of Theorem 1.1, which needs more than $\varepsilon n$ good blocks so that a polynomial missing a label in each misses more than $\varepsilon n$ labels.