Fractal projections with an application in number theory

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