TITLE: The digit sum: $s(na)=s(nb)$
QUESTION [7 upvotes]: Not that I was serious about the following question, but I think it is a must-to-ask as a follow-up to this MO post. 
For integer $n\ge0$, let $s(n)$ denote the sum of the digits in the decimal representation of $n$.

Is it true that for any integer $a,b>0$, the ratio of which is not a power of $10$, the set of all those $n\ge 0$ with $s(an)=s(bn)$ has zero density?

REPLY [7 votes]: You can find some explicit asymptotics in a paper by J. Schmidt, "The joint distribution of the binary digits of integer multiples". He considers base $2$, and theorem 1.2 says that for any $a,b$ odd and arbitrary integer $k$.
$$|\lbrace 0\le n\le x: s(na)-s(nb)=k\rbrace|=\frac{x}{\sqrt{2\pi V\cdot \log_2x }}e^{-\frac{k^2}{2V\log_2 x}}+O(\frac{x}{\log x}),$$
where $V=\frac{1}{2}\left(1-\frac{\gcd^2(a,b)}{ab}\right)$. Something similar should work for any base.