From Dimension Drop to Aperiodic Order
Abstract
We consider the parametrised family of sets E(x,y)={∑n=1∞εn/4n: (εn) ∈ {0,x,y,x+y}ℕ} for (x,y) ∈ ℕ2. This family can be viewed through three lenses: (a) as homogeneous self-similar sets; (b) as achievement sets of bi-geometric series; or (c) as the set of `rational' orthogonal projections of the four-corner Cantor set. We synthesise these three perspectives to obtain a complete topological classification of E(x,y) for (x,y) ∈ ℕ2. Next, we collapse this topological classification to a binary one according to whether or not E(x,y) has interior. When this binary classification is visualised, it reveals a two-colour tiling T of the lattice ℕ2, which, despite being visibly structured, turns out to be aperiodic; indeed, we prove it has no non-trivial translational symmetries. Due to the rigidity of our model, this same binary classification simultaneously captures several dichotomies. Most notably, when the family {E(x,y)}(x,y) ∈ ℕ2 is viewed through the theory of self-similar sets, T can be seen to describe the emergence of dimension drop within the family. Finally we examine the mechanism underlying the tiling's aperiodic order. By considering the number-theoretic properties of the tiling, we characterise its substitution structure, and discover that T is a factor of a substitution tiling on four ``hidden'' arithmetically defined states.
Related
- Failure of dimension drop for harmonic measure on Ahlfors regular boundaries of dimension close to $1$
- Aperiodic maximisers of Lyapunov dimension: A counterexample to the unrestricted form of Eden's conjecture
- Failure of dimension drop for harmonic measure on Ahlfors regular flat boundaries
- Cartan Subalgebras of Dimension Drop Interval Algebras and Their Inductive Limits
- Strict Hausdorff-dimension drop for Quartic Salem Bernoulli convolutions
BibTeX
@misc{jurga2026from,
title = {{From Dimension Drop to Aperiodic Order}},
author = {Natalia Jurga and Dmytro Karvatskyi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.24623},
eprint = {2609.24623},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.24623},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - From Dimension Drop to Aperiodic Order
AU - Natalia Jurga
AU - Dmytro Karvatskyi
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.24623
UR - https://arxiv.org/abs/2609.24623
AB - We consider the parametrised family of sets E(x,y)={∑_n=1^∞ε_n/4^n: (ε_n) ∈ {0,x,y,x+y}^ℕ} for (x,y) ∈ ℕ^2. This family can be viewed through three lenses: (a) as homogeneous self-similar sets; (b) as achievement sets of bi-geometric series; or (c) as the set of `rational' orthogonal projections of the four-corner Cantor set. We synthesise these three perspectives to obtain a complete topological classification of E(x,y) for (x,y) ∈ ℕ^2. Next, we collapse this topological classification to a binary one according to whether or not E(x,y) has interior. When this binary classification is visualised, it reveals a two-colour tiling T of the lattice ℕ^2, which, despite being visibly structured, turns out to be aperiodic; indeed, we prove it has no non-trivial translational symmetries. Due to the rigidity of our model, this same binary classification simultaneously captures several dichotomies. Most notably, when the family {E(x,y)}_(x,y) ∈ ℕ^2 is viewed through the theory of self-similar sets, T can be seen to describe the emergence of dimension drop within the family. Finally we examine the mechanism underlying the tiling's aperiodic order. By considering the number-theoretic properties of the tiling, we characterise its substitution structure, and discover that T is a factor of a substitution tiling on four ``hidden'' arithmetically defined states.
ER -