For example, as an important special case there is the analytical 0-divisors conjecture: Let T be a self-adjoint element of the complex group ring of a free group of $l^2$-norm at most 1. Consider the sequence $t_n$ of complex numbers: $t_n$ is the coefficient of the neutral element of the element $(1-T)^n$ of the group ring (so this is a combinatorial thing.)
Theorem (P. Linnel): If T is not 0 then the limit of the sequence $t_n$ is 0.