A famous conjecture of Erdős states that if S is a subset of the positive integers
and the sum of the reciprocals of elements of S diverges, then S contains arbitrarily
long arithmetic progressions. If one could prove, for each positive integer k,
sufficiently good bounds for the size of the largest subset of the first N integers lacking
k-term arithmetic progressions, then Erdős’s conjecture would follow. There is thus
great interest in the problem of proving the strongest possible bounds for sets lacking
arithmetic progressions of a fixed length. In this talk, I will survey the recent advances
of Bloom–Sisask on this problem for length three progressions and of Croot–Lev–Pach
and Ellenberg–Gijswijt on the analogous problem in F^n_3 (the "cap set problem"). These
two advances rely on very different techniques —Fourier analytic methods and a
version of the polynomial method, respectively— and I will give an overview of both.
