Let $D$ be a subset of the decimal digits $ \{ 0, 1, 2, \ldots, 9\}$, with $D \neq \{0\}$ or $\emptyset$. Let $N$ be the set of positive integers whose decimal representations (without leading $0$'s) consist only of digits in $D$. Let $R$ be the set of ratios of numbers in $N$: $R = \{ {x \over y}: x, y \in N\}$. What are the density properties of $R$ in the positive reals?
Clearly, if $D$ contains only one digit, the only accumulation points of $R$ are the powers of $10$ (as well as $0$ and $\infty$, if you consider the compactification of the positive reals), and if $D = \{0,\ldots,9\}$, $R$ is the positive rationals and is dense in the positive reals. Numerical evidence suggests that if $D = \{1,2,3,4\}$, $R$ is dense in the positive reals.
For what sets $D$ is $R$ dense in the positive reals? Generalize for all integral bases $\ge 2$.