Abstract
We introduce the notion of tiling spaces for metric spaces. The class of tiling spaces contains the Euclidean spaces, the middle-third Cantor set, and various self-similar spaces appearing in fractal geometry. For doubling tiling spaces, we characterize metric subspaces whose Assouad dimension coincides with that of the whole space.
Highlights
Fraser and Yu [3] provided a characterization of subsets of a Euclidean space whose Assouad dimension coincides with that of the whole space
They proved in [3] that for every subset F of the N-dimensional Euclidean space RN, the following are equivalent: (1) F asymptotically contains arbitrary large arithmetic patches; (2) F satisfies the asymptotic Steinhaus property; (3) dimA F = N, where dimA stands for the Assouad dimension; (4) CdimA F = N, where CdimA stands for the conformal Assouad dimension; (5) F has a weak tangent with non-empty interior; (6) the closed unit ball B(0, 1) in RN is a weak tangent to F
They used this characterization to study the problem of whether specific subsets of Euclidean spaces related to number theory, such as the products of the set of all prime numbers, asymptotically contain higher dimensional arithmetic progression
Summary
Fraser and Yu [3] provided a characterization of subsets of a Euclidean space whose Assouad dimension coincides with that of the whole space. There exists a pre-tiling space failing the condition (U) whose tiles have infinite similarity classes (see Example 6.2). If a tiling space (X, P ) satisfies the assumption that the conformal dimensions of all the tiles of (X, P ) and X are equal to dimA X, the condition that CdimA F = dimA X is equivalent to the conditions (1)–(5) stated in Theorem 1.1. For N ≥ 2 and s ∈ (0, 1), let S be an (N, s)-similar iterated function system on a complete metric space with the strong open set condition.
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