Pisot substitution sequences, one dimensional cut-and-project sets and bounded remainder sets with fractal boundary

Abstract

This paper uses a connection between bounded remainder sets in ${\mathbb{R}}^d$ and cut-and-project sets in $\mathbb{R}$ together with the fact that each one-dimensional Pisot substitution sequence is bounded distance equivalent to some lattice in order to construct several bounded remainder sets with fractal boundary. Moreover it is shown that there are cut-and-project sets being not bounded distance equivalent to each other even if they are locally indistinguishable, more precisely: even if they are contained in the same hull.

16 pages, 3 figures. Added references and polished formulations and presentation in V2

References in corpus (1)

• Pattern Equivariant Cohomology and Theorems of Kesten and Oren

Cited by in corpus (4)

• A dichotomy for bounded displacement equivalence of Delone sets
• Multiscale Substitution Tilings
• Number of bounded distance equivalence classes in hulls of repetitive Delone sets
• Balancedness and coboundaries in symbolic systems