User yui nishizawa - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T23:55:26Z http://mathoverflow.net/feeds/user/14934 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/64211/roths-theorem-and-behrends-lower-bound Roth's theorem and Behrend's lower bound Yui Nishizawa 2011-05-07T16:33:21Z 2011-09-30T01:05:45Z <p>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</p> <p>$CN(\log\log N)^5/\log N \ge r_3(N) \ge N\exp(-D\sqrt{\log N})$</p> <p>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?</p> <p>Thank you.</p> http://mathoverflow.net/questions/64211/roths-theorem-and-behrends-lower-bound Comment by Yui Nishizawa Yui Nishizawa 2011-05-17T20:45:01Z 2011-05-17T20:45:01Z Thanks to all for your responses!