  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.
