I'm not sure if this question is appropriate or even making sense, but I still feel curious: why are every example of divergent Fourier series SO COMPLICATED? It usually takes pages to construct and then prove its divergence, and when we approach the end of the proof we totally lose the insight of the construction, and it then becomes extremely difficult to even follow the formal deduction. Why? Many theorems' proofs are hard but it's clear what is being thought in this step and next, but for this one it doesn't seem to be the case! It is because there might be some easy divergent Fourier series to be discovered, or it can be proven that there's no easy divergent Fourier series? Or maybe the proof of the existence is much easier than the explicit construction? Or even maybe we still do not know much about the general theory?(The last one seems to be the least possible.)
(And I shall be asked "what does the OP mean by being 'hard'? If the OP says this is hard, what word is he using for Carleson's theorem?" This is why I stated at the beginning that I feel this question might be unclear, and I apologize if anyone feels offended, yet I posted this question anyway.)