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If I understand your comments below, this is actually false, you didn't have any expectations about the bounds for this problem. Presumably this claim was just AI generated.
The improvement in the paper goes to n(log n)^(1-(2^-182)), which while kind of neat is essentially the same as n log n (unless we're multiplying numbers on a computer larger than the universe). It's a toy result. Your amazements suggests you probably didn't even look at the paper.
In other words, I consider your thoughts on this totally untrustworthy. Please don't use AI to write your posts.
Ahem. Please look at this comment I had left a scant few minutes back.
https://www.themotte.org/post/3960/culture-war-roundup-for-the-week/485779?context=8#context
I am well aware that this is practically indistinguishable from O(n log n). I know how exponents work. I haven't checked yet, but a sufficiently ridiculous constant factor would make it even more impractical. That's been known to happen.
I had read mathematicians, on Twitter, expressing surprise that we went below the previous SOTA by any margin, no matter how minuscule. The model rephrased my notice of secondhand surprise into a first person version.
That is such an innocuous, irrelevant change that I wouldn't have bothered to correct it for it's own sake. I hadn't even told the model to only wrap my references in links. It was entirely at liberty to add minor context. The only reason I point it out is because I respect @2rafa, and wanted to declare it as a concrete example of a phrase I hadn't hand-typed.
Totally untrustworthy? What a massive overreaction. A totally untrustworthy person would deny the whole thing. Instead, I'm punished for admitting any use. You're lucky I think the price of honesty is acceptable.
If you write me off, that's your problem rather than mine. Particularly since I didn't use AI to "write" my post, I had already written a post, and I threw into it for minor improvements and explicit citations I didn't have the time to source. By my standards, the changes I've quoted from the original are positive or at least benign.
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If for the longest time the best exponent was some nice number like an integer or rational with small denominator, and the algorithm reaching it relatively simple, then even the tiniest improvement is notable. That it never performs better in cases small enough to be encourted in the real world, is not what mathematicians care about. Were it so, Big O notation would not be as common, and ultrafinistism would be the majority position.
I don't think I agree with that perspective on Big O notation. It's common because generally people don't intuitively understand the difference between, say, quadratic and exponential growth. But of course in computing it's crucial, so people need to be taught it over and over so that they will understand it intuitively. It has a huge impact on how fast an algorithm performs in real world implementations.
The difference here is that the smallest example where log(n)^(1-2^-182) is noticeably different is if you were multiplying two numbers whose physical representation in bits is larger than the universe. It's totally meaningless.
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The best exponent was some nice number because those are easier for humans to develop proofs for.
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