Character values for a finite group are sums of nth roots of unity. I'm wondering if there are any results bounding nonzero values of irreducible characters away from zero. Or if not are there examples of natural sequences of groups where entries in the character table can be made arbitrarily close zero. I don't see anything like this in Isaacs book for example.


**Added:** Thanks for the answers below, I of course should have figured out the direct product once you have one of norm <1. I guess the problem is more interesting for simple groups, for example the alternating groups the nonzero values are bounded away from zero.



Character values bounded away from zero
Although as has been observed, $|\chi(x)|$ can be positive but arbitrarily close to zero, I think it is interesting that whenever $\chi(x)$ is not a root of unity or zero, then $|\chi(y)| > 1$ for some element $y$ that generates the same cyclic subgroup as $x$.


