Projections of self-affine fractals

We extend Falconer's 1988 landmark result on the dimensions of self-affine fractals to encompass the dimensions of their projections, showing furthermore that their families of exceptional projections contain algebraic varieties which are preserved by the underlying linear algebraic group.

The techniques which we develop allow us to construct examples of additional new phenomena: firstly, we give general examples of equilibrium measures on self-affine fractals which admit non-exact-dimensional projections.

Secondly, we construct strongly irreducible self-affine sets which have small sumsets without any arithmetic resonance in their construction.