A breakthrough on integer complexity by Konyagin and Oganesyan!

So joshuazelinsky recently sent me a link to this paper and wow! This paper contains just two theorems but both are huge advances in integer complexity; one on the upper bound, one on the lower bound. First the upper bound. Let's review -- what upper bounds are known on integer complexity? If one wants a bound that works for all n, well, there's the naive bound ||n||&le;3log 2 n, and then there's…