This talk is on dimension theory for sets arising from iterated function systems, with a particular emphasis on self-affine sets. In 1988, Falconer proved that, for a fixed collection of matrices, the Hausdorff dimension of the corresponding self-affine set is something known as the affinity dimension, for Lebesgue almost every choice of translation vectors. I discuss an orthogonal approach, introducing a class of affine iterated function systems in which, given translation vectors, for Lebesgue almost all matrices, the dimension of the corresponding self-affine set is the affinity dimension. The talk will focus on describing and motivating the results. Time premitting, I will also say a few words about the proofs, which rely on Ledrappier-Young theory that was recently verified for affine iterated function systems by Barany and Kaenmaki, and a new transversality condition.

The work is joint with Balazs Barany and Antti Kaenmaki.