> I have not actually read the paper. In fact, there are three closely related papers [TC26a], [TC26b], [TC26c], and I have not read any of them. I tried, but soon gave up. The math(s) dump does not appear to be strong on exposition. Instead, I asked Fable to explain [TC26a] to me. Fable seems to understand the argument and thinks that it is correct.

this does not feel like a healthy place to be

> The new result does not directly imply new complexity bounds for polynomial multiplication over a fixed finite field, say multiplication in F2[x]. The state of the art for this problem is an algorithm that multiplies polynomials of degree n in time O(n log n 4^log*(n)) [HvdH19].

@platypii and I obtained a Lean proof of the same improvement for DFT over prime-order finite fields (yielding polynomial multiplication via the convolution theorem): https://github.com/platypii/integer-mult-bounds-lean/blob/ma...

Note in this case the title says "O(n log n)", but the important part is this is "o(n log n)", not just "O(n log n)'. The former is a strict subset of the latter. We already knew how to multiply two n-bit integers in O(n log n) time, via the FFT.

Roughly speaking, O(n log n) means "grows no faster than k(n log n) for some k", whereas o(n log n) means grows strictly more slowly* than k(n log n) for all* choices of k.

For those of us uninformed, what’s the difference between O and o?

Knuth originally defining them https://dl.acm.org/doi/10.1145/1008328.1008329

f(n) = O(g(n)) means as n increases, f(n)/g(n) must not grow beyond a constant

f(n) = o(g(n)) means as n increases, f(n)/g(n) must shrink

Nitpick:

s/must shrink/must eventually shrink, after finitely many n/

Equivalently, must go to zero in the limit.

that's not the convention I know... (from CLRS algorithms book).

Big O is sort of like <=. little o is sort of like <.

https://dl.acm.org/doi/10.1145/1008328.1008329

yes, GP is slightly off, but the idea of <= and < is accurate.

O(n) is: "there exists a K such that, for sufficiently large n, K*n is always larger".

o(n) is: "for every K, for sufficiently large n, K*n is always larger". (So o(2n+1) and o(2n) and o(n) are the same thing.)

> Lim f/g -> C a constant.

This isn't quite correct. O(g) doesn't require the limit to exist.

I do hope you can find and reattach your ass; that does sound painful.

Both are upper bounds. But small o is strictly smaller than upper bound. Big O is not greater than upper bound.

https://stackoverflow.com/questions/1364444/difference-betwe...

ty, fixed. (ahh, copy paste.)

Awesome write up and intro by a fellow Sydney-sider, kept me hooked! As someone who's not a maths wizard, the technical details flew by my head but the prose allowing me to empathise deeply with those in the field was very enlightening. I never realised how big of a deal the maths dump was and how much it has shaken up the field. I guess we are all just trying to figure out this whole "intelligence on tap" thing together.

You don’t need to be a math wizard. Just upload the paper to an AI and ask, “Explain this like I’m five.” Then, with each follow-up prompt, increase the age and level of expertise you want it to assume.