I thought it was entertainingly specific, and fortunately the abstract at least [0] was slightly less immediately beyond me than I feared, to satisfy my curiosity a bit:
The main proof is an improvement to the 4^k bound (standing since 1935) to (4-eps)^k for some epsilon.
They additionally prove (I guess they have properties that make it slightly easier than other rounder/bigger numbers?) it for eps=2^-10 and eps=2^-7 specifically.
(3.993 then comes from the latter, 4 minus it gives 3.9921 and change, but of course you need to round that up to 3.993 in order to say it's a bound: it's not definitely less than 3.992, since it could lie between the two.)
So yes maybe/it probably can be improved from 3.993, because that's a bit of a tangential claim anyway - the main thing is that it's 'some non-zero amount less than 4'^k.
(But mostly yes it was beyond me, I won't pretend to be able to even attempt to understand the proof really.)
The main proof is an improvement to the 4^k bound (standing since 1935) to (4-eps)^k for some epsilon.
They additionally prove (I guess they have properties that make it slightly easier than other rounder/bigger numbers?) it for eps=2^-10 and eps=2^-7 specifically.
(3.993 then comes from the latter, 4 minus it gives 3.9921 and change, but of course you need to round that up to 3.993 in order to say it's a bound: it's not definitely less than 3.992, since it could lie between the two.)
So yes maybe/it probably can be improved from 3.993, because that's a bit of a tangential claim anyway - the main thing is that it's 'some non-zero amount less than 4'^k.
(But mostly yes it was beyond me, I won't pretend to be able to even attempt to understand the proof really.)
[0]: https://arxiv.org/abs/2303.09521