Some properties of the binary expressions of $3^n$ are known, e.g. here was proven that the only periodic expression happens at $n=1$, or here it is shown that the number of $1$s in the expression goes to infinty.
Performing some numerical simulations it seems that zeroes and ones are quite "uniformly distributed". In particular it seems that the length $L_n$ of the longest sequence of consecutive ones in the binary expression of $3^n$ is much smaller than $n$: for $n\le 10000$ the longest sequence is made of $24$ ones.
My question is: are some inequalities like $$L_n\le c_\alpha n^\alpha, \quad\text{for some }\; 0<\alpha<1$$ known? (if not stronger ones.)
