Universiteit Utrecht

Department of Mathematics

Beta expansions of minimal weight

Wolfgang Steiner
(LIAFA, Paris)

13 February 2007

We consider digital expansions of real numbers and integers which are minimal with respect to the absolute sum of digits. It is well known that every integer  N  has an expansion in base  2  with digits  −1,0,1  such that no two consecutive non-zero digits occur and that the weight of this Non-Adjacent Form (NAF) is minimal among all expansions in base  2  of  N.  It is not difficult to determine all expansions of minimal weight in an integer base. For the Fibonacci numeration system, a particular expansion of minimal weight was given by Heuberger (2004).

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. If beta is the root of the polynomial  X 2 − X − 1  (the golden mean) or  X 3 − X 2 − X − 1  (the Tribonacci number), this automaton can be given explicitely. For particular classes of expansions of minimal weight, which are similar to the NAF, the sequence of digits can be represented by a Markov chain, which allows calculating the asymptotic probability of a digit.

The results can be extended to expansions of integers in the Fibonacci and the Tribonacci numeration systems.

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Yuri Yakubovich