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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 eachone-dimensional Pisot substitution sequence is bounded distance equivalent tosome lattice in order to construct several bounded remainder sets with fractalboundary. Moreover it is shown that there are cut-and-project sets being notbounded distance equivalent to each other even if they are locallyindistinguishable, more precisely: even if they are contained in the same hull.

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