Fractal projections with an application in number theory

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 showed that for each integer $k,$ there is a number $M>0$ such that if $b_1,\dots,b_k$ are multiplicatively independent integers greater than $M$, there are infinitely many integers whose base $b_1,b_2,\dots,b_k$ expansions all do not have any zero digits.