We calculate the growth rate of the complexity function for polytopal cut and project sets. This generalizes work of Julien where the almost canonical condition is assumed. The analysis of polytopal cut and project sets has often relied on being able to replace acceptance domains of patterns by so-called cut regions. Our results correct mistakes in the literature where these two notions are incorrectly identified. One may only relate acceptance domains and cut regions when additional conditions on the cut and project set hold. We find a natural condition, called the quasicanonical condition, guaranteeing this property and demonstrate by counterexample that the almost canonical condition is not sufficient for this. We also discuss the relevance of this condition for the current techniques used to study the algebraic topology of polytopal cut and project sets.

In this article pattern statistics of typical cubical cut and project sets are studied. We give estimates for the rate of convergence of appearances of patches to their asymptotic frequencies. We also give bounds for repetitivity and repulsivity functions. The proofs use ideas and tools developed in discrepancy theory.

We explore the problem of finding the Hausdorff dimension of the set of points that recur to shrinking targets on a self-affine fractal. To be exact, we study the dimension of a certain related symbolic recurrence set. In many cases, this set is equivalent to the recurring set on the fractal.

We establish a connection between gaps problems in Diophantine approximation and the frequency spectrum of patches in cut and project sets with special windows. Our theorems provide bounds for the number of distinct frequencies of patches of size r, which depend on the precise cut and project sets being used, and which are almost always less than a power of log r. Furthermore, for a substantial collection of cut and project sets we show that the number of frequencies of patches of size r remains bounded as r tends to infinity. The latter result applies to a collection of cut and project sets of full Hausdorff dimension.

We study tangent sets of strictly self-affine sets in the plane. If a set in this class satisfies the strong separation condition and projects to a line segment for sufficiently many directions, then for each generic point there exists a rotation ${\mathcal{O}}$ such that all tangent sets at that point are either of the form ${\mathcal{O}}((\mathbb{R}\times C)\cap B(0,1))$ , where $C$ is a closed porous set, or of the form ${\mathcal{O}}((\ell \times \{0\})\cap B(0,1))$ , where $\ell$ is an interval.

We study the dimension of code tree fractals, a class of fractals generated by a set of iterated function systems. We first consider deterministic affine code tree fractals, extending to the code tree fractal setting the classical result of Falconer and Solomyak on the Hausdorff dimension of self-affine fractals generated by a single iterated function system. We then calculate the almost sure Hausdorff, packing and box counting dimensions of a general class of random affine planar code tree fractals. The set of probability measures describing the randomness includes natural measures in random $V$-variable and homogeneous Markov constructions.