Upper and lower estimates for integer complexity

Let $\|n\|$ stand for the integer complexity of the number $n$, i.e. for the least number of $1$'s needed to write $n$ using arbitrary many additions, multiplications, and parentheses. The two-sided inequality $3\log_3 n\leq\|n\|\leq 3\log_2 n$ for all $n$ is well known and reveals the logarithmic behaviour of the complexity function $\|n\|$. While the lower bound $3\log_3 n$ is attained infinitely many times at powers of $3$, the best upper estimate is still unknown, although there are some improvements of the trivial bound $3\log_2 n$. Besides, for $``$typical$"$ numbers, i.e. for almost all numbers $n$, the better inequality $\|n\|\leq C_{avg}\log n$ holds, where, importantly, $C_{avg}\approx 3.236<\sup_{n} \frac{\|n\|}{\log n}$. We show that in fact $\|n\|\leq C_{avg}\log n+o(\log n)$ as $n\to\infty$, which, in particular, yields that $\limsup\limits_{n\to\infty}\frac{\|n\|}{\log n}\leq C_{avg}$. We also obtain the first nontrivial lower bound $\|n\|\geq 3.06\log_3 n$ for almost all numbers $n$.