k-pseudorandom measures - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T04:28:34Z http://mathoverflow.net/feeds/question/7071 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/7071/k-pseudorandom-measures k-pseudorandom measures Thomas Bloom 2009-11-28T19:26:49Z 2011-05-25T20:11:48Z <p>In reading the paper of Green and Tao on arithmetic progressions within the primes, I became very interested in the notion of a k-pseudorandom measure discussed in that paper. </p> <p>A measure here is a function $\nu:\mathbf{Z}_N\to\mathbf{R}$ such that $\mathbf{E}\nu=1+o(1)$, and it is k-pseudorandom if it obeys the ($k2^{k-1}$,$3k-1$,$k$) (I think) linear forms condition, which basically asserts that it behaves independently with respect to at most $k2^{k-1}$ independent linear forms in $3k-1$ variables, and if it also obeys the correlation condition, which is a weaker form controlling the linear forms $x+h_i$.</p> <p>They show that a relative Szemeredi's theorem applies to functions bounded by a k-pseudorandom measure, and then construct one that (effectively) bounds the primes.</p> <p>My question is where else these type of functions have been studied, whether their theory has been expanded, and whether other explicit examples have been found and applied in other situations.</p> http://mathoverflow.net/questions/7071/k-pseudorandom-measures/7077#7077 Answer by Boris Bukh for k-pseudorandom measures Boris Bukh 2009-11-28T20:40:46Z 2009-12-02T00:26:52Z <p>Linear forms condition says that these functions are morally the functions that are close to $1$ in appropriate $U^k$ norm. What I mean is that $U^k$ norms are a special kind of linear forms, and so linear forms condition implies proximity to $1$ in $U^k$, on one hand. On the other hand,if one controls $\nu-1$ in $U^t$ norm for sufficiently large $t=t(k)$, then by Cauchy-Schwarz argument one can control arbitrary linear forms.</p> <p>[EDIT: The rest of the answer is result of my misunderstanding. See the comments.] There is an unpublished work of David Conlon and Timothy Gowers on Szemerédi-type results in random sets, in which, if I understood correctly what David explained to me, they show as a special case that the control in an appropriate $U^t$ norm is enough. (In particular the correlation condition is no longer necessary, and was an artifact of the original proof.)</p> <p>So, the answer to your question is that the theory of these functions is essentially the theory of functions with small $U^k$ norm.</p>