Signed $\beta$-expansions of minimal weight

Wolfgang Steiner

Université Paris VII, CNRS, France

Coauthor(s): Christiane Frougny

We consider $\beta$-expansions on signed digit sets which are minimal with respect to the absolute sum of digits. It is well known that every integer has a signed binary expansion without two consecutive non-zero digits, and that the weight of this Non-Adjacent Form (NAF) is minimal among all signed binary expansions. Heuberger (2004) showed that there exists a kind of NAF providing minimal weight expansions for the Fibonacci numeration system as well. In his form, every digit $a\in\{-1,1\}$ is followed either by $000$ or $00(-a)$.

In this talk, we focus on expansions of real numbers in a real base $\beta$. For a certain class of bases, which are all Pisot numbers, we can prove that all expansions of minimal weight are given by a finite automaton. We have determined this automaton explicitely for the golden mean and the Tribonacci number. This allows exhibiting particular expansions of a simple form and calculating the average number of non-zero digits. A similar automaton provides all minimal weight expansions of integers in the Fibonacci and the Tribonacci numeration systems.