I ought to get something serious done over the break, so I would like to read through the proof of the theorem about primes in arithmetic progressions. But it seems it might make more sense to start with a proof of Szemeredi’s Theorem. I worked through a proof of Roth’s Theorem last year for a reading course on analytic number theory, and that wasn’t too hard. Is there a proof of Szemeredi’s Theorem that is accessible? Gowers has a proof of it on his website, but his proof is nearly three times the length of Szemeredi’s original paper. Still it apparently generalizes the technique Roth used to proof the simpler case of three-term arithmetic progressions.
I suppose it’s not that unreasonable for these theorems to have long and difficult proofs given that even the proof of van der Waerden’s Theorem, if not really all that hard, is still pretty involved. Erdos apparently hoped that Szemeredi’s Theorem would be the deeper result behind van der Waerden’s Theorem that would give a nice easy proof, but it doesn’t seem to be the case. The still more powerful conjecture of Erdos, that if S is a subset of the positive integers so that the sum of the reciprocals of the elements of S diverges, then S contains arbitrarily long arithmetic progressions, remains unsolved.