Arithmetics in Generalised Cantor Base Systems

For alternate Cantor real base numeration systems we generalise the result of Frougny and Solomyak on arithmetics on the set of numbers with finite expansions. We generalise the notion of finiteness property, denoted ( F ). We also provide several necessary conditions, and comment on a sufficient condition of this property. The sufficient condition allows us to find a set of generalised Cantor bases with Property ( F ). The used construction also provides a method for performing addition of finite expansions in Cantor real bases.