Title: Minimal characteristic factors in arithmetic combinatorics
Speaker: Julia Wolf
Speaker Info: University of Cambridge and Institute for Advanced Study
Let L be a system of linear equations. It follows from the work of Host and Kra that any ergodic measure preserving system has a factor characteristic for the multiple ergodic average associated with L that is isomorphic to an inverse limit of d-step nilsystems for some d. Since every nilmanifold has a natural tower of factors, one is lead to ask for the degree of the minimal characteristic factor for L. This question was completely resolved by a recent preprint of Leibman.Date: Tuesday, October 30, 2007
We will present an analogous result in the context of arithmetic combinatorics, which was obtained independently of Leibman's work in collaboration with Tim Gowers. Given a linear system L, we determine the minimal degree of uniformity required of a subset A of F_p^n in order to guarantee that A contains roughly the expected number of solutions to L, that is, the number of solutions we would expect to obtain in a random subset of F_p^n of the same density as A.
While uniformity of degree 1 is a classical concept that can be interpreted in a purely Fourier analytic sense, higher-degree uniformity norms first appeared in the work of Gowers on Szemeredi's Theorem for progressions of length k. They have since found numerous applications, notably in the work of Green and Tao on long progressions in the primes.
The talk will include plenty of examples illustrating the rich analogies between ergodic and number theory, and shall assume no prior knowledge of results in arithmetic combinatorics