Tao 2026 local bernstein theory lower bounds lebesgue

../

corollary_1_11: For any triangular array of distinct nodes and any omega(n) tending to infinity, a dense set of points where the Lebesgue function is at least (2/pi) log n - omega(n) for infinitely many n.

evidence/: Retains the independent review of the bounded Theorem 1.10(i) transfer and the E1153 source corrections.

theorem_1_10_i_transfer: Records Tao v3’s exact local bound and proves its elementary transfer to E1153.

theorem_1_10_ii: Tao's lower bound (4|I|/pi^2) log n - o(log n) for the integral of the Lebesgue function over a fixed interval, uniformly in distinct nodes in [-1,1], with 8/pi^2 on the whole interval.

theorem_1_13: Sharp sup-norm and integral lower bounds, 2 and 8, for the sum of |P(x)|/|P'(x_k)| over the 2n distinct zeroes of a degree-n trigonometric polynomial on [0, 2pi).

theorem_1_6: Tao's local Bernstein, Boas and Duffin--Schaeffer inequalities and local zero count for a function holomorphic on a rectangle, real and bounded on its lower edge, with errors depending on the distance to the vertical sides.

---

Terence Tao, Local Bernstein theory, and lower bounds for Lebesgue constants, arXiv:2603.21453v3.

Version and source scope

The copy read for this card is the 51-page v3 PDF. It has an arXiv version stamp of 22 April 2026; the title-page author date separately says 23 April 2026. The arXiv API response read identifies v3, updated 2026-04-22T03:16:58Z, with the comment “51 pages, 12 figures. Further corrections”; its first-submission timestamp is 2026-03-23T00:15:40Z. This is a preprint. No publisher acceptance or complete independent proof review is established by those metadata. The arXiv record names arXiv's non-exclusive distribution license (arXiv:2603.21453), every other right reserved.

Selected mathematical content

The paper develops local Bernstein theory for functions holomorphic on a rectangle $\{x+iy:x\in I,\ 0\le y\le y_0\}$ that are real and bounded on its bottom edge, of at most exponential size on its top edge and of at most double-exponential size on its vertical sides. Theorem 1.6 (p. 4) is the local result: local Bernstein, Boas and Duffin--Schaeffer inequalities and a local zero count, the counterparts of parts (i)--(iv) of the global Theorem 1.4 (pp. 2--3).

For ordinary Lagrange interpolation at arbitrary distinct nodes in $[-1,1]$, Theorem 1.10(i), p. 9, proves

$$\sup_{x\in I}\sum_k|l_k(x)|\ge\frac2\pi\log n-O_I(1)$$

on each fixed positive-length interval $I\subseteq[-1,1]$, for sufficiently large $n$ depending on $I$. The constant is uniform in the nodes. The exact statement and elementary transfer explain continuity, supremum versus maximum, and the strict $-o(1)$ form in Problem 1153. The underlying source proof is accessible, but its complete proof has not been compiled and independently reviewed here.

Part (ii) gives $\int_I\lambda\ge(4|I|/\pi^2)\log n-o(\log n)$, hence the full-interval coefficient $8/\pi^2$. This is weaker in its error term than the conjectured lower bound (1.23), which includes a constant and $o(1)$. Corollary 1.11 gives the separate dense-set, fixed-point, infinite-subsequence bound with any divergent positive loss $\omega(n)$. It is related to Problem 1132, without settling its stronger questions. The method uses residue representations, local Bernstein inequalities, and control at macroscopic, mesoscopic and microscopic scales. Theorem 1.13 (p. 10) is the sharp trigonometric toy model of both parts of Theorem 1.10.

Source-declared AI provenance

The author’s disclosure is on p. 14, §1.7. It credits ChatGPT DeepResearch, Gemini DeepResearch and Claude for locating references; ChatGPT Pro, Gemini Pro and Claude for suggestions on which the proof of Lemma 2.5 is based; and GPT, which Nat Sothanaphan, Aron Bhalla and Tao each used on their own to find corrections in an earlier manuscript. It credits AlphaEvolve and ChatGPT Pro for discovery of the Theorem 1.13 proof via Lemma 1.14, GitHub Copilot for text completion, Claude Code for routine typesetting, and Gemini for most plots. Apart from these uses, the author describes the text as human-written. Footnote 7 on p. 12 separately credits Nat Sothanaphan using GPT for a rigorously proved version of a claim reproduced as Theorem 4.1(i). Footnote 8 on p. 16 credits ChatGPT Pro with the argument for Lemma 1.14(i), and footnote 10 on p. 19 says an initial version of the proof of Lemma 2.5 was provided by ChatGPT. These are the author’s disclosures, not independent correctness findings or formal-verification evidence.

Versioned discussion record

The E1153 discussion thread distinguishes Nat Sothanaphan’s 27 February 2026 report of a proposed proof gap in an earlier manuscript from Tao’s 24 March announcement of a complete rewrite on arXiv. Later on 24 March Sothanaphan reported four likely easy fixes, and Tao said they would be addressed in the next revision. The selected v3 is dated 22 April. No comment-by-comment comparison with v3 was performed here; the February report is not evidence that the same gap remains in v3. The statement and proof-review limits above govern this home.

Bears on.

• Problem 1153: Theorem 1.10(i) bounds the supremum of the Lebesgue function over a fixed interval below by $\frac2\pi\log n-O(1)$, uniformly in distinct nodes; the elementary transfer on that page turns this into the problem's maximum over $[a,b]$ exceeding $(\frac2\pi-o(1))\log n$.
• Problem 1132: Corollary 1.11 gives, for any triangular array of distinct nodes and any $\omega(n)\to\infty$, a dense set of points with $\lambda^{(n)}\ge\frac2\pi\log n-\omega(n)$ for infinitely many $n$; it answers neither the problem's constant-loss question nor its almost-everywhere question.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.