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][1]. 

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? 

[1]: https://mathoverflow.net/questions/94525/equal-digit-sums