Let $a_n=10^n \cdot \pi$. Is the set of numbers $\{a_n-\lfloor a_n \rfloor : n \in \mathbb{N}\}$ dense in [0,1]?

What is the best known result near this question?

Apparently John Nash asked this on an undergraduate analysis exam (according to an anecdote told by Seymour Haber, recounted in Sylvia Nassar's biography of Nash).

non-lacunary. An example is`$S=\{2^m3^n\mid m,n\ge 0\]$`

. An example of a lacunary $S$ (so the result does not apply) is`$S=\{10^n\mid n\ge0\}$`

. $\endgroup$3more comments