The most general theorem for PAC learning of Boolean functions that I am aware of is the theorem in section 3.4 of Ryan O'Donnel's book where its basically shown that Boolean functions whose Fourier weights are concentrated on some modes can be PAC learnt in polynomial time. Also it trivially follows that in $2^n$ queries one can learn any Boolean function on the $n-$Hamming cube.
My question is two fold,
For what concept classes do we know of high ($2^n$?) query lowerbound for PAC learning? If the proof is short then it would be very helpful if you could kindly type it in or give a reference.
For Boolean functions whose Fourier spectrum seems very evenly spread out over the modes it seems that it should be hard to learn them in a few queries. Is this intuition right and if yes then what is the most quantitative theorem we know of which captures this?
I am interchangeably using the words "query complexity" and "running time of the learning algorithm". I hope this is okay. If not then it would be helpful if you could point out the subtlety that I am missing.