The following question emerged from thinking about the Erdős discrepancy problem. I don't know whether an answer would be directly helpful, but it might, and in any case I find the question quite a nice one.

Suppose we have a character on the additive group of the rationals. That is, we have a function $\chi$ from $\mathbb{Q}$ to $\mathbb{T}$ such that $\chi(r+s)=\chi(r)\chi(s)$ for any two rational numbers $r$ and $s$. The following argument shows that for every $\epsilon$ there exists a positive integer $n$ such that $|\chi(n^{-1})-1|<\epsilon$. You choose a positive integer $r$ that's bigger than $2\pi/\epsilon$, then set $N=r!$, and let $\alpha$ be such that $\chi(1/N)=e(\alpha)$ (which is standard notation for $\exp(2\pi i\alpha)$). Then you use Dirichlet's pigeonhole argument to find $n\leq r$ such that the distance from $n\alpha$ to the nearest integer is at most $1/r$ and therefore less than $\epsilon/2\pi$. From that it follows that $|\chi(n/N)-1|<\epsilon$, and we are done, since $N/n$ is an integer.

This argument gives a pretty disappointing bound: $n$ has an $x^x$ type dependence on $1/\epsilon$. Can more sophisticated techniques (Kloosterman sums perhaps?) give much better bounds?