A simple condition for bounded displacement

Abstract

We study separated nets Y that come from primitive substitution tilings of the Euclidean space R d . We show that the question whether Y is a bounded displacement of Z d or not can be reduced, in most cases, to a simple question on the eigenvalues and eigenspaces of the substitution matrix.

Limit Theorems for Self-Similar Tilings

“…Solomon [32,33] gives deviation estimates similar to ours for the number of tiles in a "super-tile" of high order. The tiles are assumed to be bi-Lipschitz equivalent to a ball, but the substitution need not be non-periodic.…”

“…These deviation bounds are sharp, at least, in the general case. There are related recent results by Solomon [32,33] and Aliste-Prieto, Coronel, Gambaudo [2,3], who obtained estimates for the rate of convergence to frequency of the number of prototiles per volume for a class of domains. They were motivated by questions on bi-Lipschitz equivalence and bounded displacement of separated nets, arising from self-similar tilings, to the lattice.…”

“…It is an efficient hierarchical "packing" of a Lipschitz domain by tiles of varying orders. Analogous constructions have been used in [32,33,2,3,31].…”

“…We claim that for each u ∈ E + , (38) defines a finitely-additive measure on the algebra of subsets of Γ T i −x generated by the sets ω k (Γ ω,T j −y ), with j ≤ m, k ≥ 0 and y ∈ R d such that T i − x ∈ ω k (T j − y). It is enough to verify finite additivity in (33), since the general case reduces to it easily. We have…”

Pattern equivariant cohomology and theorems of Kesten and Oren

“…4 Herex and I are identified with their coset representatives in [0, 1). 5 See [1,16,17,18,37,38] for recent developments and [8,28,36] for some earlier developments.…”