The dimension theory of self-affine sets generated by finite iterated function systems (IFSs) has been developed since the 1980s, when it was investigated for which types of sets the Hausdorff and box-counting dimensions coincide. In 1988, Falconer introduced the affinity dimension, an expression which purely depends on the singular values of the linear parts of the affine maps in the IFS. It turns out that for finitely generated self-affine sets, the affinity dimension is an upper bound for the upper box-counting dimension and hence the Hausdorff dimension. Moreover, Falconer proved that the Hausdorff dimension is almost surely (with respect to the translation vectors of the affine maps in the IFS) equal to the minimum of the dimension of the ambient space and the affinity dimension.  Since the development of these results, several advancements deciphering when the Hausdorff dimension of self-affine sets generated by finite IFSs is equal to the minimum of the dimension of the ambient space and the affinity dimension of the IFS have been made.  Additionally, results in this direction for self-affine sets generated by infinite IFSs which are irreducible, meaning the linear parts of the affine maps do not all preserve a common proper non-trivial linear subspace, have also been developed.

In this talk we will discuss some recent results concerning how the affinity dimension relates to the Hausdorff dimension and the lower box-counting dimension for a family of self-affine sets generated by infinite non-irreducible IFSs.  We will also discuss the dimension spectrum of this family of self-affine sets, which are, in fact, related to restricted digit sets of signed Lüroth expansions.

This is joint work with Sven van Golden, Charlene Kalle and Sabrina Kombrink.