相关论文: Tilings of the infinite $p$-ary tree and Cantor ho…

In this paper, we provide a geometric characterization of tiles in the finite abelian groups \( \mathbb{Z}_{p^n} \times \mathbb{Z}_q \) and \( \mathbb{Z}_{p^n} \times \mathbb{Z}_p \) using the concept of a \( p \)-homogeneous tree, which…

We prove that for any infinite countable amenable group $G$, any $\epsilon > 0$ and any finite subset $K\subset G$, there exists a tiling (partition of $G$ into finite "tiles" using only finitely many "shapes"), where all the tiles are $(K;…

A finite collection $P$ of finite sets tiles the integers iff the integers can be expressed as a disjoint union of translates of members of $P$. We associate with such a tiling a doubly infinite sequence with entries from $P$. The set of…

We introduce a topology on the space of all isomorphism types represented in a given class of countable models, and use this topology as an aid in classifying the isomorphism types. This mixes ideas from effective descriptive set theory and…

We prove that fairly general spaces of tilings of R^d are fiber bundles over the torus T^d, with totally disconnected fiber. This was conjectured (in a weaker form) in [W3], and proved in certain cases. In fact, we show that each such space…

In this paper, we prove that given any \Pi^0_1 subset $P$ of $\{0,1\}^\NN$ there is a tileset $\tau$ with a set of configurations $C$ such that $P\times\ZZ^2$ is recursively homeomorphic to $C\setminus U$ where $U$ is a computable set of…

In the present paper, as we did previously in [7], we investigate the relations between the geometric properties of tilings and the algebraic properties of associated relational structures. Our study is motivated by the existence of…

This study explores the properties of the function which can tile the field $\mathbb{Q}_p$ of $p$-adic numbers by translation. It is established that functions capable of tiling $\mathbb{Q}_p$ is by translation uniformly locally constancy.…

Tree sets are abstract structures that can be used to model various tree-shaped objects in combinatorics. Finite tree sets can be represented by finite graph-theoretical trees. We extend this representation theory to infinite tree sets.…

Identity-homotopic self-homeomorphisms of a space of non-periodic 1-dimensional tiling are generalizations of orientation-preserving self-homeomorphisms of circles. We define the analogue of rotation numbers for such maps. In constrast to…

We introduce a notion of equivalence on tilings which is formulated in terms of their local structure. We compare it with the known concept of locally deriving one tiling from another and show that two tilings of finite type are…

Let $T$ be a tile in $\mathbb{Z}^n$, meaning a finite subset of $\mathbb{Z}^n$. It may or may not tile $\mathbb{Z}^n$, in the sense of $\mathbb{Z}^n$ having a partition into copies of $T$. However, we prove that $T$ does tile $\mathbb{Z}^d$…

This is a survey article on trees, with a modest number of proofs to give a flavor of the way these topologies can be efficiently handled. Trees are defined in set-theorist fashion as partially ordered sets in which the elements below each…

In this paper we present with algebraic trees a novel notion of (continuum) trees which generalizes countable graph-theoretic trees to (potentially) uncountable structures. For that purpose we focus on the tree structure given by the branch…

We introduce a new family of nonperiodic tilings, based on a substitution rule that generalizes the pinwheel tiling of Conway and Radin. In each tiling the tiles are similar to a single triangular prototile. In a countable number of cases,…

Any bounded tile of the field $\mathbb{Q}_p$ of $p$-adic numbers is a compact open set up to a zero Haar measure set. In this note, we give a simple and direct proof of this fact.

This paper demonstrates that every ultrametric space is homeomorphic to a clade space of a pruned tree, i.e., a subspace of a tree's canopy. Furthermore, it characterizes several topological properties of ultrametrizable spaces through the…

We formalize an existing computability-theoretic method of presenting first-order structures whose domains have the cardinality of the continuum. Work using these methods until now has emphasized their topological properties. We shift the…

Let $\mathcal{T}^{(p)}_n$ be the set of $p$-ary labeled trees on $\{1,2,\dots,n\}$. A maximal decreasing subtree of an $p$-ary labeled tree is defined by the maximal $p$-ary subtree from the root with all edges being decreasing. In this…

It is proved that whenever two aperiodic repetitive tilings with finite local complexity have homeomorphic tiling spaces, their associated complexity functions are asymptotically equivalent in a certain sense (which implies, if the…