Projections of self-affine fractals

TL;DR

This paper extends Falconer's results to the projections of self-affine fractals, revealing new phenomena such as non-exact-dimensional projections and special sumset properties.

Contribution

It generalizes the dimension theory of self-affine fractals to their projections and introduces novel examples with unique geometric and algebraic features.

Findings

Projections of self-affine fractals can have non-exact dimensions.

Families of exceptional projections form algebraic varieties.

Constructs self-affine sets with small sumsets and no arithmetic resonance.

Abstract

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.