Definition and fundamentals of tilings generated by Pisot numbers are shown by arithmetic consideration. Results include the case that a Pisot number does not have a finitely expansible property, i.e. a sofic Pisot case. Especially we show that the boundary of each tile has Lebesgue measure zero under some weak condition.
"On the boundary of self affine tilings generated by Pisot numbers." J. Math. Soc. Japan 54 (2) 283 - 308, April, 2002. https://doi.org/10.2969/jmsj/05420283