On the number of conjugacy classes of zeros of characters

Letm be a fixed non-negative integer. In this work we try to answer the following question: What can be said about a (finite) groupG if all of its irreducible (complex) characters vanish on at mostm conjugacy classes? The classical result of Burnside about zeros of characters says thatG is abelian ifm=0, so it is reasonable to expect that the structure ofG will somehow reflect the fact that the irreducible characters vanish on a bounded number of classes. The same question can also be posed under the weaker hypothesis thatsome irreducible character ofG hasm classes of zeros. For nilpotent groups we shall prove that the order is bounded by a function ofm in the first case but only the derived length can be bounded in general under the weaker condition. For solvable groups the situation is not so well understood although we shall prove that the Fitting height can be bounded by a double logarithmic function ofm, improving a recent result by G. Qian.