Members' Colloquium

Several of the most important problems in combinatorial number theory ask for the size of the largest subset of some abelian group or interval of integers lacking points in some arithmetic configuration. One example of such a question is "What is the largest subset of $\{1,\cdots,N\}$ with no nontrivial $k$-term arithmetic progressions $x,x+y,...,x+(k-1)y$?". Gowers initiated the study of higher order Fourier analysis while seeking to answer this question and used it to give the first reasonable quantitative bounds.