A number theorist I know (who studies Galois representations) raised a question recently:


  Which finite groups can have an irreducible character of degree at least 2 having only $n=2, 3$, or 4 classes where the character takes nonzero values? 


He has learned about a few special examples involving nonabelian groups which are very close to being abelian.    My limited intuition about the question, based on finite groups of Lie type, suggests that the sort of group he is looking for will be far from simple.   But maybe there is no reasonable characterization  of these groups for a given $n \leq 4$?   In any case, there should be some relevant literature out there.   The question itself belongs to a familiar genre in finite group theory:  What does the character table tell me about the structure of a group? 
Appropriate references: S. , Gagola, Pacific J. Math 108 (1983), 363-385,

Berkovich-Zhmud', Characters of Finite Groups, Part 2, Chapter 21.

For vector spaces, $\dim (U + V) = \dim U + \dim V - \dim (U \cap V)$, so
$$
\dim(U +V + W) = \dim U + \dim V + \dim W - \dim (U \cap V) - \dim (U \cap W) - \dim (V \cap W) + \dim(U \cap V \cap W),
$$
right?