Roth's theorem on 3-term arithmetic progressions (3AP) is concerned with the value of $r\_3(N)$, which is defined as the cardinality of the largest subset of the integers between 1 and N with no non-trivial 3AP. The best results as far as I know are that


$CN(\log\log N)^5/\log N \ge r\_3(N) \ge N\exp(-D\sqrt{\log N})$


for some constants $C,D>0$. The upper bound is by Tom Sanders in 2010 and the lower bound is by Felix Behrend dating back to 1946. My question is this: even though the upper and lower bounds are still quite a bit apart, I hear mathematicians hinting that something close to Behrend's lower bound might be giving the correct order, such as in Ben Green's paper "Roth's theorem in the primes", and I am curious as to where this is coming from. Is it because there has been no significant improvement on the lower bound (whereas there has been lot's of work on the upper bound)? Or in analogy to a similar type of problem? Or maybe just a casual remark? Or some other reason?


Thank you.



Roth's theorem and Behrend's lower bound
Dear Yui,


It's only slightly more than a casual remark. Our inability to find a better example is certainly a big reason for believing that Behrend's bound is correct. Julia Wolf and I slightly rehashed the proof of Behrend's bound


<http://arxiv.org/abs/0810.0732>


When formulated this way, I think the construction looks both fairly natural and fairly unimprovable.


Also, there are beginning to be hints as to the correct behaviour coming from apparently similar equations such as $x\_1 + x\_2 + x\_3 = 3x\_4$. I think that Schoen and others, using work of Sanders, may have improved the bounds for this equation to $N \exp(-\log^c N)$, though I'm not certain about this.


Despite these remarks it is not known whether better bounds for Roth's theorem follow from any other more "natural" conjectures, such as the Polynomial Freiman-Ruzsa conjecture, so any suggestion that Behrend is sharp is somewhat tenuous. Some other people think differently, I believe - that it may not be sharp.


