I was wondering: I find this topic interesting from a conceptual point of view, but I'm reluctant to learn more about it without having some concrete useful application as motivation. Can anyone give an example like that?
The simplest example would be testing how close f is to a linear function.
We consider f:{0,1}^n->{0,1}. f is linear if f(x+y)=f(x)+f(y), and there are different ways to measure almost linearity. Either the probability that randomly picking x and y, we have f(x+y)=f(x)+f(y), the other would be how far away the function is from a linear function in hamming distance.
Using the tools in analysis of boolean functions, we can show these two measures are basically the same.