Does anyone know of any lower bounds on the number of nonzero digits that appear in powers of 2 when written to base 3? (Other than the easy "If it's more than 8 it has to have at least 3.") I know there's been some stuff done with powers of 2 written to base 3, but I can't seem to find anything that quite answers this question.


(Are there more general such bounds? Powers of a written to base b? (To avoid triviality, suppose that no power of a is a power of b... or are stricter conditions needed?) From what I can tell, it looks like even specific instances of these are hard, so I suppose I should stick to the specific version.)



Lower bound on # of nonzero digits in ternary expansions of powers of 2?
A nontrivial lower bound can be found in a paper of Cam Stewart (see <http://www.math.uwaterloo.ca/PM_Dept/Homepages/Stewart/Jour_Books/J-reine-ange-Math-1980.pdf>). He proves, more generally, for fixed bases a and b for which $\log a/\log b$ is irrational, that the sum of the number of nonzero digits in the base a and base b digits of an integer n exceeds (essentially) $$\log \log n/ \log \log \log n.$$


