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Some linear dynamical systems over finite fields are studied and the self-similar character of their development is proved. Connections with aperiodic tilings, Delanoy numbers and other topics are also proved. The prime fields F_p have a…
We prove that the upper box dimension of an inhomogeneous self-affine set is bounded above by the maximum of the affinity dimension and the dimension of the condensation set. In addition, we determine sufficient conditions for this upper…
We show that in the hierarchical tile assembly model, if there is a producible assembly that overlaps a nontrivial translation of itself consistently (i.e., the pattern of tile types in the overlap region is identical in both translations),…
This paper investigates the dimension theory of some families of continuous piecewise linear iterated function systems. For one family, we show that the Hausdorff dimension of the attractor is equal to the exponential growth rate obtained…
We study Bedford--McMullen type carpets whose selected grid rectangles may be reflected in one or both coordinates. The organizing principle is that the Hausdorff dimension is controlled by the entropy of the weak-coordinate projection.…
This paper extends some results of [M5] and [M3], in particular, removing assumptions of positive lower density. We give conditions on a general family $P_{\lambda}:\mathbb{R}^{n}\to\mathbb{R}^{m}, \lambda \in \Lambda,$ of orthogonal…
For each object in a tensor triangulated category, we construct a natural continuous map from the object's support---a closed subset of the category's triangular spectrum---to the Zariski spectrum of a certain commutative ring of…
We study the Hausdorff dimension of self-similar sets and measures on the line. We show that if the dimension is smaller than the minimum of 1 and the similarity dimension, then at small scales there are super-exponentially close cylinders.…
We study subsets of $\R^{d}$ which are thin for doubling measures or isotropic doubling measures. We show that any subset of $\R^{d}$ with Hausdorff dimension less than or equal to $d-1$ is thin for isotropic doubling measures. We also…
We study the topology and the Hausdorff dimension of a random Cantor set with overlaps, generated by an iterated function system with scaling ratio equal to the Golden Mean. The results extend known formulas to a case where the Open Set…
We study the dimension theory of limit sets of iterated function systems consisting of a countably infinite number of contractions. Our primary focus is on the intermediate dimensions: a family of dimensions depending on a parameter $\theta…
The method of alternating projections (MAP) is a common method for solving feasibility problems. While employed traditionally to subspaces or to convex sets, little was known about the behavior of the MAP in the nonconvex case until 2009,…
We prove that maximal annuli in $\mathbb{L}^{3}$ bounded by circles, straight lines or cone points in a pair of parallel spacelike planes are part of either a Lorentzian catenoid or a Lorentzian Riemann's example. We show that under the…
We calculate the Assouad and lower dimensions of graph-directed Bedford-McMullen carpets, which reflect the extreme local scaling laws of the sets, in contrasting with known results on Hausdorff and box dimensions. We also investigate the…
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.…
We consider a generalisation of the self-affine iterated function systems of Lalley and Gatzouras by allowing for a countable infinity of non-conformal contractions. It is shown that the Hausdorff dimension of the limit set is equal to the…
We obtain several results for (iterated) planar contact manifolds in higher dimensions: (1) Iterated planar contact manifolds are not weakly symplectically semi-fillable. This generalizes a 3-dimensional result of Etnyre to a…
In this paper we propose a new model of random graph directed fractals that extends the current well-known model of random graph directed iterated function systems, $V$-variable attractors, and fractal and Mandelbrot percolation. We study…
In this article, we study the Hausdor dimension of weakly conformal IFS's shrinking targets with possible overlaps, provided the conformality dimension of the systems and the dimension of the attractor are equal. Those results extends the…
The set of chambers of a real hyperplane arrangement may be ordered by separation from some fixed chamber. When this poset is a lattice, Bjorner, Edelman, and Ziegler proved that the chambers are in natural bijection with the biconvex sets…