Abstract

Given an affine-linear form L in t variables with integer coefficients, a subset A of [N]={1,2,...,N} is said to be L-free if A^t does not contain any (non-trivial) solution of the equation L(x)=0. The greatest cardinality that an L-free subset of [N] can have is denoted r_L(N).

I will discuss recent joint work with Olof Sisask which proves the convergence of r_L(N)/N (and of other related quantities) as N tends to infinity, for any given form L in at least 3 variables. The proof uses the discrete Fourier transform and tools from arithmetic combinatorics. The convergence result addresses a question of Imre Ruzsa and extends work of Ernie Croot.

In the different context where intervals [N] are replaced by cyclic groups of prime order, we have similar convergence results, and I will discuss how in this context the limits can be related to natural analogous quantities defined on the circle group.