Proportion of $0,1,-1$s in the values of a rational-valued irreducible character

Question

In a finite group $G$, is there a lower bound for how many times $\chi(g)$ can take the values $0$ or $\pm 1$?

Here, I'm considering $\chi$ irreducible and rational valued.

Answer 1

I have never seen this particular question before, so maybe someone else will know whether there is a lower bound. There certainly isn't for zeros in irreducible characters, the proportion of which can take any value in $[0,1]$. The best I can do is to prove that any lower bound is at most $1/6$.

I haven't checked this completely, but the group given by $$\langle a,b,c,d,e,f|a^2=b^2=c^2=d^3=e^3=f^p=1,b^a=bc,d^a=d^b=d^2,e^b=e^2,f^a=f^{-1}\rangle$$ (where all other conjugations are trivial, i.e., $f^b=f$, etc. should have a character, the inflation of the 2-dimensional character for the $D_8$ quotient by $\langle d,e,f\rangle$, where the proportion tends to $1/6$ as $p$ grows. For $p=7$ the proportion is 13/51=0.255, $p=11$ gives $17/75=0.227$, $p=29$ gives $35/183=0.191$, and so on.

Since this character takes values $0,2,-2$, it's just those that have value $0$ that we need to consider. I haven't counted the number of such classes, but there are clearly three of orders $2$ or $4$. There should be two of order $6$, two of order $12$, and $p-1$ of order $2p$, so $p+6$ such classes in total. I haven't computed exactly the number of classes, but it's around $6p$.