Let $b$ be a fixed integer $\ge 2$ and $A$ a proper subset of $\{0, \ldots, b-1\}$. Then define $X$ to be the set of all positive integers whose base-$b$ representation consists only of digits from $A$. It's known that the series $\sum_{x \in X} \frac{1}{x}$ is convergent, see the article on Kempner series on Wiki.en and references therein, which, together with a classical result from additive-theory folklore, implies that the natural density of $X$ is zero. But what is known, say, about the upper Banach density of $X$? I seem to have a proof that this is still zero, yet would like to understand if the result is buried somewhere in the literature (but not too deep, so that someone here around can give a reference) and/or follows from relatively well-known facts.

Added later. Just in case it may be useful, let me recall that, given $S \subseteq \mathbf N$, we take the natural (or asymptotic) density of $S$ to be the limit: $$\lim_{n \to \infty} \frac{|X \cap [1,n]|}{n},$$ whenever this exists, and the upper Banach (or uniform) density of $X$ to be the limit: $$\lim_{n \to \infty} \max_{m \ge 1} \frac{|X \cap [m,m+n]|}{n},$$ which always exists (by Fekete's lemma on subadditive real sequences).