In this paper, we discuss a connection between geometric measure theory and number theory. This method brings a new point of view for some number-theoretic problems concerning digit expansions. Among other results, we show that for each integer k, there is a number $M>0$ such that if $b_{1},\ldots ,b_{k}$ are multiplicatively independent integers greater than M, there are infinitely many integers whose base $b_{1},b_{2},\ldots ,b_{k}$ expansions all do not have any zero digits.