Local Bernstein theory, and lower bounds for Lebesgue constants
terrytao.wordpress.com
terrytao.wordpress.com
> 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.
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.)It would be surreal if a carpentry forum was regularly abuzz about mountaineering because climbers use a hammer-shaped tool.
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.
Anyway, there are a lot of topics in the discussion in HN and many times someone can give some insight. Sometimes it's about compilers and there are a few users that know about that. Sometimes it's about rockets and there are a few users that know about that. Sometimes it's about steel alloys and there are a few users that know about that. ... Not everyone here is a programmer.
I'm using very little AI, but my wife has been using it a lot. We both agree that it's at the level of a Gold medal in the IMO, that means that it can kick our ass hard. Anyway, sometimes the AI hallucinates and sometimes makes stupid error, so it's necessary to verify the output.
[1] It looks short, so perhaps a week is enough, but I feel I'm being too optimistic. Let's say a month just to be safe.
[2] Protip: If you see a circle, take the Fourier Transform and cross your fingers. You can't believe how many problems it solves. If it's useful, add me as a co-author.