We show that the Hausdorff dimension of any slice of the graph of the Takagi function is bounded above by the Assouad dimension of the graph minus one, and that the bound is sharp. The result is deduced from a statement on more general self-affine sets, which is of independent interest. We also prove that Marstrand’s slicing theorem on the graph of the Takagi function extends to all slices if and only if the upper pointwise dimension of every projection of the length measure on the x-axis lifted to the graph is at least one.

In this paper, we study the dimension of planar self-affine sets, of which generating iterated function system (IFS) contains non-invertible affine mappings. We show that under a certain separation condition the dimension equals to the affinity dimension for a typical choice of the linear-parts of the non-invertible mappings, furthermore, we show that the dimension is strictly smaller than the affinity dimension for certain choices of parameters.

Motivated by near-identical graphs of two increasing continuous functions—one related to Zaremba’s conjecture and the other due to Salem—we provide an explicit connection between fractals and regular sequences by showing that the graphs of ghost distributions, the distribution functions of measures associated to regular sequences, are sections of self-affine sets. Additionally, we provide a sufficient condition for such measures to be purely singular continuous. As a corollary, and analogous to Salem’s strictly increasing singular continuous function, we show that the ghost distributions of the Zaremba sequences are singular continuous.

We prove that certain families of homogenous affine iterated function systems in $\mathbb {R}^{d}$ have the property that the open set condition and the existence of exact overlaps both occur densely in the space of translation parameters. These examples demonstrate that in the theorems of Falconer and Jordan–Pollicott–Simon on the almost sure dimensions of self-affine sets and measures, the set of exceptional translation parameters can be a dense set. The proof combines results from the literature on self-affine tilings of $\mathbb {R}^{d}$ with an adaptation of a classic argument of Erdős on the singularity of certain Bernoulli convolutions. This result encompasses a one-dimensional example due to Kenyon which arises as a special case.

A classical theorem of Hutchinson asserts that if an iterated function system acts on $\mathbb {R}^{d}$ by similitudes and satisfies the open set condition then it admits a unique self-similar measure with Hausdorff dimension equal to the dimension of the attractor. In the class of measures on the attractor, which arise as the projections of shift-invariant measures on the coding space, this self-similar measure is the unique measure of maximal dimension. In the context of affine iterated function systems it is known that there may be multiple shift-invariant measures of maximal dimension if the linear parts of the affinities share a common invariant subspace, or more generally if they preserve a finite union of proper subspaces of $\mathbb {R}^{d}$ . In this paper we give an example where multiple invariant measures of maximal dimension exist even though the linear parts of the affinities do not preserve a finite union of proper subspaces.