## Combinatorial and arithmetical properties of infinite words associated with non-simple quadratic Parry numbers

Doppler Institute for Mathematical Physics and Applied Mathematics and Department of Mathematics, FNSPE, Czech Technical University, Trojanova 13, 120 00 Praha 2, Czech Republic; l.balkova@centrum.cz; masakova@km1.fjfi.cvut.cz; oturek@centrum.cz

Received:
29
August
2006

Accepted:
4
January
2007

We study some arithmetical and combinatorial properties of
*β*-integers for *β* being the larger root of the equation
*x ^{2} = mx - n,m,n ∈ ℵ, m ≥ n +2 ≥ 3*. We determine with
the accuracy of

*±*1 the maximal number of

*β*-fractional positions, which may arise as a result of addition of two

*β*-integers. For the infinite word

*u*coding distances between the consecutive

_{β}>*β*-integers, we determine precisely also the balance. The word

*u*is the only fixed point of the morphism

_{β}>*A*→

*A*and

^{m-1}B*B*→

*A*. In the case

^{m-n-1}B*n = 1*, the corresponding infinite word

*u*is sturmian, and, therefore,

_{β}>*1*-balanced. On the simplest non-sturmian example with

*n*

*≥*2, we illustrate how closely the balance and the arithmetical properties of

*β*-integers are related.

Mathematics Subject Classification: 68R15 / 11A63

Key words: Balance property / arithmetics / beta-expansions / infinite words

*© EDP Sciences, 2007*