Numbers omitting digits in certain base expansions

In DOI:10.1017/etds.2022.2 the author proved that for each integer $k$ there is an implicit number $M>0$ such that if $b_1,\cdots ,b_k$ are multiplicatively independent integers greater than $M$, there are infinitely many integers whose base $b_1,b_2,\cdots ,b_k$ expansions all do not have zero digits. In this paper we don’t require the multiplicative independence condition and make the result quantitative, getting an explicit value for $M$. We also obtain a result for the case when the missing digit(s) may not be zero. Finally, we extend our method to study various missing-digit sets in an algebraic setting.