Local Bernstein theory, and lower bounds for Lebesgue constants

https://terrytao.wordpress.com/2026/03/23/local-bernstein-theory-and-lower-bounds-for-lebesgue-constants/

Comments

_alternator_Mar 27, 2026, 2:09 PM
Fascinating discussion including showing how LLMs are helping push the state of the art:

> I still did not see how to prove this inequality, but I decided to try my luck giving it to ChatGPT Pro, which recognized it as an {L^1} approximation problem and gave me a duality-based proof (based ultimately on the Fourier expansion of the square wave). With some further discussion, I was able to adapt this proof to functions of global exponential type (replacing the Fourier manipulations with contour shifting arguments, in the spirit of the Paley-Wiener theorem), which roughly speaking gave me half of what I needed to establish (2).

> As a side note, this latter argument was provided to me by ChatGPT, as I was not previously aware of the Nevanlinna two-constant theorem.

This mirrors my experience with these tools for math. Great for local problems and chatting through issues. Still can’t do the whole thing in one shot but getting there.

tensegristMar 27, 2026, 1:59 PM
Presumably one reason this is of interest here:

    The proof proceeds by a modification of the Duffin–Schaeffer argument, together with the two-constant theorem of Nevanlinna (and some standard estimates of harmonic measures on rectangles) to deal with the effect of the localization. (As a side note, this latter argument was provided to me by ChatGPT, as I was not previously aware of the Nevanlinna two-constant theorem.)
chromacityMar 27, 2026, 2:15 PM
Which I find funny in a way: I'm sure that almost no one here understands the article, grasps the significance of this problem in mathematics, or can meaningfully comment on the difficulty of solving it. But we'll still have opinions because the article mentions a popular tool some of us like, some of us dislike, and some are ambivalent about.

It would be surreal if a carpentry forum was regularly abuzz about mountaineering because climbers use a hammer-shaped tool.

TimorousBestieMar 27, 2026, 3:25 PM
Nevanlinna theory isn’t that obscure (in the sense of mathematics, I suppose) but it is very difficult (for me, probably less so for Tao) when working to have the whole of 21st century analysis in your head at once and see what could be applied where. I can see how an LLM would be quicker than a human at recognizing a context where a theorem from an apparently unrelated subfield could be applied.
adampunkMar 27, 2026, 2:53 PM
It is deeply hilarious to watch software engineers become, at long last, bikeshedders.
btillyMar 27, 2026, 3:18 PM
What do you mean, "at long last"?

We've always been bikeshedders. For example, back in Slashdot days, some company would decide to migrate something from Windows to Linux. Immediately the debate became whether they should have gone with Debian or SuSE instead of Red Hat.

adampunkMar 27, 2026, 2:55 PM
Think of this as an algebraic jewelers loupe. Zoom in on local structure and push aside the global. It’s fascinating.