Department of Mathematics

Beta expansions of minimal weight

(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