Not that I'm serious about this answer, but yes. Take an $N$ digit number $n$ and split its decimal representation into pieces of length $3M$ where $M$ is large but fixed. A noticeable portion of these pieces will be of the kind $M$ zeroes, something, $M$ zeroes. $M$ zeroes will control the transfers if $M$ is large enough and, conditioning upon the location of these pieces and the rest of the representation, we see that these pieces will give us long sums of independent variables. Now, we need 3 statements:

1) If $X$ is an integer-valued random variable with finitely many values that is not constant, then $\lim_{N\to\infty}\sup_{k\in\mathbb Z} P\{\sum_{i=1}^N X_i=k\}=0$ where $X_i$ are i.i.d. random variables equidistributed with $X$.

2) If $s(an)-s(bn)=C$ for all $n$ (i.e., if we cannot choose $M$ to use Claim 1), then $C=0$.

3) If $s(an)=s(bn)$ for all $n$, then $a$ and $b$ differ only by the number of zeroes in the end. 

Claim 3 is the only non-obvious one. I do not have a decent proof of it yet (if I find one while I'm driving to work, I'll post it, but, most likely, someone else will beat me to that).