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

TL;DR

This paper explores the relationship between Pisot substitution sequences, cut-and-project sets, and bounded remainder sets with fractal boundaries, revealing new constructions and distinctions among these mathematical objects.

Contribution

It introduces methods to construct bounded remainder sets with fractal boundaries using Pisot sequences and cut-and-project sets, and demonstrates non-equivalence of certain sets in the same hull.

Findings

Constructed bounded remainder sets with fractal boundaries.

Showed cut-and-project sets can differ even if locally indistinguishable.

Linked bounded remainder sets to Pisot substitution sequences.

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.