Erdős Problem #1153 — Local lower bounds for Lebesgue functions

Erdős and Turán asked whether arbitrary interpolation nodes in $[-1,1]$ force the Lebesgue function to reach the sharp asymptotic size on every fixed subinterval. The answer is affirmative.

Known results

• Bernstein, 1931: the full interval $[-1,1]$ has a logarithmic lower bound.
• Erdős: globally, $\max_{x\in[-1,1]}\lambda(x)\geq\frac{2}{\pi}\log n-O(1)$; Chebyshev nodes show sharpness up to $O(1)$.
• Erdős and Szabados, 1978: for every fixed $[a,b]\subseteq[-1,1]$, the restricted maximum is $\gg\log n$, without the sharp constant.

March 2026 resolution

Tao's result proves that for every fixed nontrivial interval $I\subset[-1,1]$ and arbitrary distinct nodes, $\sup_{x\in I}\lambda(x)\geq\frac{2}{\pi}\log n-o(\log n)$. This directly proves Problem #1153 and also establishes an integral lower bound over $I$.

Current status (as of March 2026): The problem is resolved affirmatively by a published arXiv theorem, with the sharp leading constant $\frac{2}{\pi}$; no substantive part remains open.